Theorem xpeq0r 4969

Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)

Assertion

Ref	Expression
xpeq0r	|- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) )

Proof of Theorem xpeq0r

Step	Hyp	Ref	Expression
1		xpeq1 4561	. . 3 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
2		0xp 4627	. . 3 |- ( (/) X. B ) = (/)
3	1, 2	eqtrdi 2189	. 2 |- ( A = (/) -> ( A X. B ) = (/) )
4		xpeq2 4562	. . 3 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
5		xp0 4966	. . 3 |- ( A X. (/) ) = (/)
6	4, 5	eqtrdi 2189	. 2 |- ( B = (/) -> ( A X. B ) = (/) )
7	3, 6	jaoi 706	1 |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1332   (/)c0 3368   X. cxp 4545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555

This theorem is referenced by:  sqxpeq0  4970