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Theorem dfpprod2 32529
 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 32502 . 2 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2 df-txp 32501 . 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 2850 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 1658  Vcvv 3415   ∩ cin 3798   × cxp 5341  ◡ccnv 5342   ↾ cres 5345   ∘ ccom 5347  1st c1st 7427  2nd c2nd 7428   ⊗ ctxp 32477  pprodcpprod 32478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-ext 2804 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-cleq 2819  df-txp 32501  df-pprod 32502 This theorem is referenced by:  pprodcnveq  32530
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