Theorem xpdisj1 4958

Description: Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Assertion

Ref	Expression
xpdisj1	|- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Proof of Theorem xpdisj1

Step	Hyp	Ref	Expression
1		inxp 4668	. 2 |- ( ( A X. C ) i^i ( B X. D ) ) = ( ( A i^i B ) X. ( C i^i D ) )
2		xpeq1 4548	. . 3 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = ( (/) X. ( C i^i D ) ) )
3		0xp 4614	. . 3 |- ( (/) X. ( C i^i D ) ) = (/)
4	2, 3	syl6eq 2186	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = (/) )
5	1, 4	syl5eq 2182	1 |- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1331   i^i cin 3065   (/)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-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-opab 3985  df-xp 4540  df-rel 4541

This theorem is referenced by:  djudisj  4961  xp01disjl  6324  xpfi  6811  djuinr  6941