Theorem xpeq1 4790

Description: Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
xpeq1	|- ( A = B -> ( A X. C ) = ( B X. C ) )

Proof of Theorem xpeq1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2302	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 469	. . 3 |- ( A = B -> ( ( x e. A /\ y e. C ) <-> ( x e. B /\ y e. C ) ) )
3	2	opabbidv 4198	. 2 |- ( A = B -> { <. x , y >. | ( x e. A /\ y e. C ) } = { <. x , y >. | ( x e. B /\ y e. C ) } )
4		df-xp 4782	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
5		df-xp 4782	. 2 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
6	3, 4, 5	3eqtr4g 2296	1 |- ( A = B -> ( A X. C ) = ( B X. C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {copab 4192   X. cxp 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-opab 4194  df-xp 4782

This theorem is used by:  xpeq12  4795  xpeq1i  4796  xpeq1d  4799  opthprc  4828  reseq2  5060  xpeq0r  5213  xpdisj1  5215  xpima1  5237  pmvalg  6937  xpsneng  7124  xpcomeng  7130  xpdom2g  7134  xpfi  7243  exmidomni  7488  exmidfodomrlemim  7559  hashxp  11289  txuni2  15482  txbas  15484  txopn  15491  txrest  15502  txdis  15503  txdis1cn  15504  xmettxlem  15735  xmettx  15736  dvmptid  15942  incistruhgr  16536