Theorem dfpprod2 35846

Description: Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)

Assertion

Ref	Expression
dfpprod2	⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

Proof of Theorem dfpprod2

Step	Hyp	Ref	Expression
1		df-pprod 35819	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		df-txp 35818	. 2 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))
3	1, 2	eqtri 2768	1 ⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

Syntax hints:   = wceq 1537  Vcvv 3488   ∩ cin 3975   × cxp 5698  ^◡ccnv 5699   ↾ cres 5702   ∘ ccom 5704  1^st c1st 8028  2^nd c2nd 8029   ⊗ ctxp 35794  pprodcpprod 35795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-txp 35818  df-pprod 35819

This theorem is referenced by:  pprodcnveq  35847