Theorem xpeq0r 4956

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 4548	. . 3 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
2		0xp 4614	. . 3 |- ( (/) X. B ) = (/)
3	1, 2	syl6eq 2186	. 2 |- ( A = (/) -> ( A X. B ) = (/) )
4		xpeq2 4549	. . 3 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
5		xp0 4953	. . 3 |- ( A X. (/) ) = (/)
6	4, 5	syl6eq 2186	. 2 |- ( B = (/) -> ( A X. B ) = (/) )
7	3, 6	jaoi 705	1 |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) )

Syntax hints:   -> wi 4   \/ wo 697   = wceq 1331   (/)c0 3358   X. cxp 4532

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542

This theorem is referenced by:  sqxpeq0  4957