Theorem dfpprod2 36736

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 36709	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		df-txp 36708	. 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 2783	1 ⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450   ∩ cin 4089   × cxp 5648  ^◡ccnv 5649   ↾ cres 5652   ∘ ccom 5654  1^st c1st 7999  2^nd c2nd 8000   ⊗ ctxp 36684  pprodcpprod 36685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-txp 36708  df-pprod 36709

This theorem is used by:  pprodcnveq  36737