Theorem txpeq1 5780

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

Assertion

Ref	Expression
txpeq1	⊢ (A = B → (A ⊗ C) = (B ⊗ C))

Proof of Theorem txpeq1

Step	Hyp	Ref	Expression
1		coeq2 4876	. . 3 ⊢ (A = B → (◡1st ∘ A) = (◡1st ∘ B))
2	1	ineq1d 3457	. 2 ⊢ (A = B → ((◡1st ∘ A) ∩ (◡2nd ∘ C)) = ((◡1st ∘ B) ∩ (◡2nd ∘ C)))
3		df-txp 5737	. 2 ⊢ (A ⊗ C) = ((◡1st ∘ A) ∩ (◡2nd ∘ C))
4		df-txp 5737	. 2 ⊢ (B ⊗ C) = ((◡1st ∘ B) ∩ (◡2nd ∘ C))
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → (A ⊗ C) = (B ⊗ C))

Syntax hints:   → wi 4   = wceq 1642   ∩ cin 3209  1^st c1st 4718   ∘ ccom 4722  ^◡ccnv 4772  2^nd c2nd 4784   ⊗ ctxp 5736

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727  df-txp 5737

This theorem is referenced by:  pprodeq1  5835