Theorem dfpprod2 34184

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 34157	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		df-txp 34156	. 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 2766	1 ⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

Syntax hints:   = wceq 1539  Vcvv 3432   ∩ cin 3886   × cxp 5587  ^◡ccnv 5588   ↾ cres 5591   ∘ ccom 5593  1^st c1st 7829  2^nd c2nd 7830   ⊗ ctxp 34132  pprodcpprod 34133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-txp 34156  df-pprod 34157

This theorem is referenced by:  pprodcnveq  34185