Theorem dfpprod2 31989

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 31962	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		df-txp 31961	. 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 2644	1 ⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

Syntax hints:   = wceq 1483  Vcvv 3200   ∩ cin 3573   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118  1^st c1st 7166  2^nd c2nd 7167   ⊗ ctxp 31937  pprodcpprod 31938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-txp 31961  df-pprod 31962

This theorem is referenced by:  pprodcnveq  31990