Theorem txpeq2 5780

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

Assertion

Ref	Expression
txpeq2	⊢ (A = B → (C ⊗ A) = (C ⊗ B))

Proof of Theorem txpeq2

Step	Hyp	Ref	Expression
1		coeq2 4875	. . 3 ⊢ (A = B → (◡2nd ∘ A) = (◡2nd ∘ B))
2	1	ineq2d 3457	. 2 ⊢ (A = B → ((◡1st ∘ C) ∩ (◡2nd ∘ A)) = ((◡1st ∘ C) ∩ (◡2nd ∘ B)))
3		df-txp 5736	. 2 ⊢ (C ⊗ A) = ((◡1st ∘ C) ∩ (◡2nd ∘ A))
4		df-txp 5736	. 2 ⊢ (C ⊗ B) = ((◡1st ∘ C) ∩ (◡2nd ∘ B))
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → (C ⊗ A) = (C ⊗ B))

Syntax hints:   → wi 4   = wceq 1642   ∩ cin 3208  1^st c1st 4717   ∘ ccom 4721  ^◡ccnv 4771  2^nd c2nd 4783   ⊗ ctxp 5735

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

This theorem is referenced by:  pprodeq2  5835