Theorem xpeq12i 4807

Description: Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)

Hypotheses

Ref	Expression
xpeq12i.1	⊢ A = B
xpeq12i.2	⊢ C = D

Assertion

Ref	Expression
xpeq12i	⊢ (A × C) = (B × D)

Proof of Theorem xpeq12i

Step	Hyp	Ref	Expression
1		xpeq12i.1	. 2 ⊢ A = B
2		xpeq12i.2	. 2 ⊢ C = D
3		xpeq12 4804	. 2 ⊢ ((A = B ∧ C = D) → (A × C) = (B × D))
4	1, 2, 3	mp2an 653	1 ⊢ (A × C) = (B × D)

Syntax hints:   = wceq 1642   × cxp 4771

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-opab 4624  df-xp 4785

This theorem is referenced by:  rnpprod  5843