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Theorem xpeq12 4641
Description: Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
xpeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Proof of Theorem xpeq12
StepHypRef Expression
1 xpeq1 4636 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2 xpeq2 4637 . 2 (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
31, 2sylan9eq 2230 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353   × cxp 4620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-opab 4062  df-xp 4628
This theorem is referenced by:  xpeq12i  4644  xpeq12d  4647  xpid11  4845  xp11m  5062  txtopon  13395  txbasval  13400  ismet  13477  isxmet  13478
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