Theorem xpeq0r 5026

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 4618	. . 3 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
2		0xp 4684	. . 3 |- ( (/) X. B ) = (/)
3	1, 2	eqtrdi 2215	. 2 |- ( A = (/) -> ( A X. B ) = (/) )
4		xpeq2 4619	. . 3 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
5		xp0 5023	. . 3 |- ( A X. (/) ) = (/)
6	4, 5	eqtrdi 2215	. 2 |- ( B = (/) -> ( A X. B ) = (/) )
7	3, 6	jaoi 706	1 |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1343   (/)c0 3409   X. cxp 4602

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  sqxpeq0  5027