[GOAL]
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f g : SpectralMap α β
h : f.toFun = g.toFun
⊢ f = g
[PROOFSTEP]
cases f
[GOAL]
case mk
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : SpectralMap α β
toFun✝ : α → β
spectral'✝ : IsSpectralMap toFun✝
h : { toFun := toFun✝, spectral' := spectral'✝ }.toFun = g.toFun
⊢ { toFun := toFun✝, spectral' := spectral'✝ } = g
[PROOFSTEP]
cases g
[GOAL]
case mk.mk
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
toFun✝¹ : α → β
spectral'✝¹ : IsSpectralMap toFun✝¹
toFun✝ : α → β
spectral'✝ : IsSpectralMap toFun✝
h : { toFun := toFun✝¹, spectral' := spectral'✝¹ }.toFun = { toFun := toFun✝, spectral' := spectral'✝ }.toFun
⊢ { toFun := toFun✝¹, spectral' := spectral'✝¹ } = { toFun := toFun✝, spectral' := spectral'✝ }
[PROOFSTEP]
congr
[GOAL]
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : SpectralMap β γ
g : SpectralMap α β
⊢ ↑f ∘ ↑g = ↑↑f ∘ ↑↑g
[PROOFSTEP]
rfl
  -- porting note: removed `simp` from this and added lemma above to address `simpNF` lint
[GOAL]
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : SpectralMap β γ
g : SpectralMap α β
⊢ ↑(comp f g) = ContinuousMap.comp ↑f ↑g
[PROOFSTEP]
rfl
[GOAL]
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : SpectralMap β γ
f₁ f₂ : SpectralMap α β
hg : Injective ↑g
h : comp g f₁ = comp g f₂
a : α
⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)
[PROOFSTEP]
rw [← comp_apply, h, comp_apply]
