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Theorem dfpprod2 34923
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
StepHypRef Expression
1 df-pprod 34896 . 2 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2 df-txp 34895 . 2 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))
31, 2eqtri 2760 1 pprod(𝐴, 𝐵) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3474  cin 3947   × cxp 5674  ccnv 5675  cres 5678  ccom 5680  1st c1st 7975  2nd c2nd 7976  ctxp 34871  pprodcpprod 34872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724  df-txp 34895  df-pprod 34896
This theorem is referenced by:  pprodcnveq  34924
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