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Theorem xpeq12 4410
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 4405 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2 xpeq2 4406 . 2 (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
31, 2sylan9eq 2135 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285   × cxp 4389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-opab 3860  df-xp 4397
This theorem is referenced by:  xpeq12i  4413  xpeq12d  4416  xpid11m  4605  xp11m  4809
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