MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpeq12i Unicode version

Theorem xpeq12i 4841
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  X.  C )  =  ( B  X.  D
)

Proof of Theorem xpeq12i
StepHypRef Expression
1 xpeq12i.1 . 2  |-  A  =  B
2 xpeq12i.2 . 2  |-  C  =  D
3 xpeq12 4838 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
41, 2, 3mp2an 654 1  |-  ( A  X.  C )  =  ( B  X.  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    X. cxp 4817
This theorem is referenced by:  xpssres  5121  imainrect  5253  cnvssrndm  5332  fpar  6390  canthwelem  8459  pjpm  16859  txbasval  17560  hausdiag  17599  ussval  18211  ex-xp  21593  ismgm  21757  ghsubgolem  21807  hh0oi  23255  isdrngo1  26264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-opab 4209  df-xp 4825
  Copyright terms: Public domain W3C validator