Theorem xpeq1 4641

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 2241	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 465	. . 3 |- ( A = B -> ( ( x e. A /\ y e. C ) <-> ( x e. B /\ y e. C ) ) )
3	2	opabbidv 4070	. 2 |- ( A = B -> { <. x , y >. | ( x e. A /\ y e. C ) } = { <. x , y >. | ( x e. B /\ y e. C ) } )
4		df-xp 4633	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
5		df-xp 4633	. 2 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
6	3, 4, 5	3eqtr4g 2235	1 |- ( A = B -> ( A X. C ) = ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {copab 4064   X. cxp 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-opab 4066  df-xp 4633

This theorem is referenced by:  xpeq12  4646  xpeq1i  4647  xpeq1d  4650  opthprc  4678  reseq2  4903  xpeq0r  5052  xpdisj1  5054  xpima1  5076  pmvalg  6659  xpsneng  6822  xpcomeng  6828  xpdom2g  6832  xpfi  6929  exmidomni  7140  exmidfodomrlemim  7200  hashxp  10806  txuni2  13759  txbas  13761  txopn  13768  txrest  13779  txdis  13780  txdis1cn  13781  xmettxlem  14012  xmettx  14013