Theorem xpeq1 4788

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 4197	. 2 |- ( A = B -> { <. x , y >. | ( x e. A /\ y e. C ) } = { <. x , y >. | ( x e. B /\ y e. C ) } )
4		df-xp 4780	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
5		df-xp 4780	. 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 4191   X. cxp 4772

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 4193  df-xp 4780

This theorem is used by:  xpeq12  4793  xpeq1i  4794  xpeq1d  4797  opthprc  4826  reseq2  5058  xpeq0r  5210  xpdisj1  5212  xpima1  5234  pmvalg  6933  xpsneng  7120  xpcomeng  7126  xpdom2g  7130  xpfi  7239  exmidomni  7483  exmidfodomrlemim  7554  hashxp  11283  txuni2  15448  txbas  15450  txopn  15457  txrest  15468  txdis  15469  txdis1cn  15470  xmettxlem  15701  xmettx  15702  dvmptid  15908  incistruhgr  16497