Theorem xpeq0r 5033

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 4625	. . 3 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
2		0xp 4691	. . 3 |- ( (/) X. B ) = (/)
3	1, 2	eqtrdi 2219	. 2 |- ( A = (/) -> ( A X. B ) = (/) )
4		xpeq2 4626	. . 3 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
5		xp0 5030	. . 3 |- ( A X. (/) ) = (/)
6	4, 5	eqtrdi 2219	. 2 |- ( B = (/) -> ( A X. B ) = (/) )
7	3, 6	jaoi 711	1 |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) )

Syntax hints:   -> wi 4   \/ wo 703   = wceq 1348   (/)c0 3414   X. cxp 4609

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619

This theorem is referenced by:  sqxpeq0  5034