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Theorem dfpprod2 32326
 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 32299 . 2 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2 df-txp 32298 . 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 2793 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 1631  Vcvv 3351   ∩ cin 3722   × cxp 5247  ◡ccnv 5248   ↾ cres 5251   ∘ ccom 5253  1st c1st 7313  2nd c2nd 7314   ⊗ ctxp 32274  pprodcpprod 32275 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764  df-txp 32298  df-pprod 32299 This theorem is referenced by:  pprodcnveq  32327
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