Theorem pprodeq2 5835

Description: Equality theorem for parallel product. (Contributed by Scott Fenton, 31-Jul-2019.)

Assertion

Ref	Expression
pprodeq2	⊢ (A = B → PProd (C, A) = PProd (C, B))

Proof of Theorem pprodeq2

Step	Hyp	Ref	Expression
1		coeq1 4874	. . 3 ⊢ (A = B → (A ∘ 2nd ) = (B ∘ 2nd ))
2		txpeq2 5780	. . 3 ⊢ ((A ∘ 2nd ) = (B ∘ 2nd ) → ((C ∘ 1st ) ⊗ (A ∘ 2nd )) = ((C ∘ 1st ) ⊗ (B ∘ 2nd )))
3	1, 2	syl 15	. 2 ⊢ (A = B → ((C ∘ 1st ) ⊗ (A ∘ 2nd )) = ((C ∘ 1st ) ⊗ (B ∘ 2nd )))
4		df-pprod 5738	. 2 ⊢ PProd (C, A) = ((C ∘ 1st ) ⊗ (A ∘ 2nd ))
5		df-pprod 5738	. 2 ⊢ PProd (C, B) = ((C ∘ 1st ) ⊗ (B ∘ 2nd ))
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → PProd (C, A) = PProd (C, B))

Syntax hints:   → wi 4   = wceq 1642  1^st c1st 4717   ∘ ccom 4721  2^nd c2nd 4783   ⊗ ctxp 5735   PProd cpprod 5737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  freceq12  6311