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Theorem pprodeq1 5834
 Description: Equality theorem for parallel product. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
pprodeq1 (A = BPProd (A, C) = PProd (B, C))

Proof of Theorem pprodeq1
StepHypRef Expression
1 coeq1 4874 . . 3 (A = B → (A 1st ) = (B 1st ))
2 txpeq1 5779 . . 3 ((A 1st ) = (B 1st ) → ((A 1st ) ⊗ (C 2nd )) = ((B 1st ) ⊗ (C 2nd )))
31, 2syl 15 . 2 (A = B → ((A 1st ) ⊗ (C 2nd )) = ((B 1st ) ⊗ (C 2nd )))
4 df-pprod 5738 . 2 PProd (A, C) = ((A 1st ) ⊗ (C 2nd ))
5 df-pprod 5738 . 2 PProd (B, C) = ((B 1st ) ⊗ (C 2nd ))
63, 4, 53eqtr4g 2410 1 (A = BPProd (A, C) = PProd (B, C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  1st c1st 4717   ∘ ccom 4721  2nd c2nd 4783   ⊗ ctxp 5735   PProd cpprod 5737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726  df-txp 5736  df-pprod 5738 This theorem is referenced by: (None)
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