NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  xpeq12 GIF version

Theorem xpeq12 4803
Description: Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
xpeq12 ((A = B C = D) → (A × C) = (B × D))

Proof of Theorem xpeq12
StepHypRef Expression
1 xpeq1 4798 . 2 (A = B → (A × C) = (B × C))
2 xpeq2 4799 . 2 (C = D → (B × C) = (B × D))
31, 2sylan9eq 2405 1 ((A = B C = D) → (A × C) = (B × D))
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358   = wceq 1642   × cxp 4770
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 4623  df-xp 4784
This theorem is referenced by:  xpeq12i  4806  xpeq12d  4809  xpid11  4926  xp11  5056  fnpprod  5843  mucnc  6131
  Copyright terms: Public domain W3C validator