Theorem xpeq0r 4746

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 4359	. . 3 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
2		0xp 4420	. . 3 |- ( (/) X. B ) = (/)
3	1, 2	syl6eq 2088	. 2 |- ( A = (/) -> ( A X. B ) = (/) )
4		xpeq2 4360	. . 3 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
5		xp0 4743	. . 3 |- ( A X. (/) ) = (/)
6	4, 5	syl6eq 2088	. 2 |- ( B = (/) -> ( A X. B ) = (/) )
7	3, 6	jaoi 636	1 |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) )

Syntax hints:   -> wi 4   \/ wo 629   = wceq 1243   (/)c0 3224   X. cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353

This theorem is referenced by: (None)