Theorem xpeq0r 5210

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 4788	. . 3 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
2		0xp 4855	. . 3 |- ( (/) X. B ) = (/)
3	1, 2	eqtrdi 2287	. 2 |- ( A = (/) -> ( A X. B ) = (/) )
4		xpeq2 4789	. . 3 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
5		xp0 5207	. . 3 |- ( A X. (/) ) = (/)
6	4, 5	eqtrdi 2287	. 2 |- ( B = (/) -> ( A X. B ) = (/) )
7	3, 6	jaoi 728	1 |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) )

This proof depends on syntax axioms:   -> wi 4   \/ wo 720   = wceq 1402   (/)c0 3520   X. cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782

This theorem is used by:  sqxpeq0  5211