| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, v} #{ x // IsAlgebraic R x } ≤ lift.{v, u} #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n ","nextTactic":"rw [← mk_uLift]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #(ULift.{u, v} { x // IsAlgebraic R x }) ≤ lift.{v, u} #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n ","nextTactic":"rw [← mk_uLift]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #(ULift.{u, v} { x // IsAlgebraic R x }) ≤ #(ULift.{v, u} R[X]) * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n ","nextTactic":"choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\n⊢ #(ULift.{u, v} { x // IsAlgebraic R x }) ≤ #(ULift.{v, u} R[X]) * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n ","nextTactic":"refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ lift.{u, v} #↑(g ⁻¹' {f}) ≤ ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n ","nextTactic":"rw [lift_le_aleph0]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ #↑(g ⁻¹' {f}) ≤ ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n ","nextTactic":"rw [le_aleph0_iff_set_countable]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ Set.Countable (g ⁻¹' {f})","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n ","nextTactic":"suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\nthis : MapsTo Subtype.val (g ⁻¹' {f}) (rootSet f A)\n⊢ Set.Countable (g ⁻¹' {f})\ncase this\nR : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ MapsTo Subtype.val (g ⁻¹' {f}) (rootSet f A)","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n ","nextTactic":"exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"case this\nR : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ MapsTo Subtype.val (g ⁻¹' {f}) (rootSet f A)","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n ","nextTactic":"rintro x (rfl : g x = f)","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"case this\nR : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nx : ↑{x | IsAlgebraic R x}\n⊢ ↑x ∈ rootSet (g x) A","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n ","nextTactic":"exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{v, u} (max #R ℵ₀) * ℵ₀ ≤ max (lift.{v, u} #R) ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by ","nextTactic":"simp","declUpToTactic":"theorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.59_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ Set.Countable {x | IsAlgebraic R x}","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n ","nextTactic":"rw [← le_aleph0_iff_set_countable]","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ #↑{x | IsAlgebraic R x} ≤ ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n ","nextTactic":"rw [← lift_le]","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ lift.{?u.51811, v} #↑{x | IsAlgebraic R x} ≤ lift.{?u.51811, v} ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n ","nextTactic":"apply (cardinal_mk_lift_le_max R A).trans","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "} | |
| {"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ max (lift.{v, u} #R) ℵ₀ ≤ lift.{u, v} ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n ","nextTactic":"simp","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "} | |
| {"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #{ x // IsAlgebraic R x } ≤ #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n ","nextTactic":"rw [← lift_id #_]","declUpToTactic":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.96_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ "} | |
| {"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n rw [← lift_id #_]\n ","nextTactic":"rw [← lift_id #R[X]]","declUpToTactic":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n rw [← lift_id #_]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.96_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ "} | |
| {"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ lift.{u, u} #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n rw [← lift_id #_]\n rw [← lift_id #R[X]]\n ","nextTactic":"exact cardinal_mk_lift_le_mul R A","declUpToTactic":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n rw [← lift_id #_]\n rw [← lift_id #R[X]]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.96_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ "} | |
| {"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #{ x // IsAlgebraic R x } ≤ max #R ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n rw [← lift_id #_]\n rw [← lift_id #R[X]]\n exact cardinal_mk_lift_le_mul R A\n#align algebraic.cardinal_mk_le_mul Algebraic.cardinal_mk_le_mul\n\ntheorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n ","nextTactic":"rw [← lift_id #_]","declUpToTactic":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.102_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ "} | |
| {"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n rw [← lift_id #_]\n rw [← lift_id #R[X]]\n exact cardinal_mk_lift_le_mul R A\n#align algebraic.cardinal_mk_le_mul Algebraic.cardinal_mk_le_mul\n\ntheorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n rw [← lift_id #_]\n ","nextTactic":"rw [← lift_id #R]","declUpToTactic":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n rw [← lift_id #_]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.102_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ "} | |
| {"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n rw [← mk_uLift]\n rw [← mk_uLift]\n choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n rw [lift_le_aleph0]\n rw [le_aleph0_iff_set_countable]\n suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n rintro x (rfl : g x = f)\n exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n (cardinal_mk_lift_le_mul R A).trans <|\n (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n rw [← le_aleph0_iff_set_countable]\n rw [← lift_le]\n apply (cardinal_mk_lift_le_max R A).trans\n simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n rw [← lift_id #_]\n rw [← lift_id #R[X]]\n exact cardinal_mk_lift_le_mul R A\n#align algebraic.cardinal_mk_le_mul Algebraic.cardinal_mk_le_mul\n\ntheorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n rw [← lift_id #_]\n rw [← lift_id #R]\n ","nextTactic":"exact cardinal_mk_lift_le_max R A","declUpToTactic":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n rw [← lift_id #_]\n rw [← lift_id #R]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.102_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ "} | |