Is low pressure compressor air used to cool turbine blades?
Short answer:

Low pressure compressor air can be used for turbine cooling, but usually it can only be used to cool the low pressure turbine. For the HPT, you need cooling air from the HPC. This is because the cooling air must come from a location in the compressor that is a higher pressure than the part of turbine it is cooling, so it will actually flow into that turbine location.

Longer answer:

Cooling air from the compressor is air that the engine has compressed, so put energy into, but doesn’t get the full energy out of, when it enters the turbine as cooling air. So, you want to use as little as possible, and take it from the early stages of the compressor, rather than the back of the compressor, where even more energy has been put into it.

But.... the cooling air has to be taken from a location within the compressor that is a higher pressure than the turbine location which it is cooling, so that it actually flows into the turbine.

Hence, to reduce the parasitic nature of cooling air, the low pressure turbine is generally cooled with air from the early stages of the compressor, but the high pressure turbine must use cooling air from a sufficiently high source, so it usually comes from the last few stages of the high pressure compressor. As explained, this is necessary for the pressures to be favorable to cause the cooling air to flow. However, aside from meaning it is more parasitic towards the engine, there is another disadvantage of having to use high pressure air for cooling. Pressuring the air causes its temperature to increase. So, the “cooling” air from the exit of the high pressure compressor is hotter than cooling air from the start of the HPC. Because of this, sometimes the cooling air is cooled by a heat exchanger in the bypass duct, because the cooling air from the exit of the HPC is actually quite hot.

Why the picture shows both LP and HP air in the same blade:

In the picture you show, the same blade is being cooled by both LP and HP air. This appears to contradict this answer. If one can use cooler LP air, why use the hotter, more parasitic HP air. I found the source of your image, and the text there doesn’t explain this contradiction. But if you look at the two images attached, you can see that the pressure around a turbine blade is not uniform. There are higher pressure regions especially at the front of the blade, while the pressure at the trailing edge is lower. Since you always need your cooling air to have a higher pressure then the static pressure at the location of your outlet, you need high pressure cooling air at the front where the pressure on the blade is high and you can get away with using lower pressure cooling air at the back since the pressure on the blades trailing edge is low.