Theorem xpdisj1 3474

Description: Cross products with disjoint sets are disjoint.

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		xpeq1 3206	. . 3 |- ((A i^i B) = (/) -> ((A i^i B) X. (C i^i D)) = ((/) X. (C i^i D)))
2		xp0r 3245	. . 3 |- ((/) X. (C i^i D)) = (/)
3	1, 2	syl6eq 1526	. 2 |- ((A i^i B) = (/) -> ((A i^i B) X. (C i^i D)) = (/))
4		inxp 3275	. 2 |- ((A X. C) i^i (B X. D)) = ((A i^i B) X. (C i^i D))
5	3, 4	syl5eq 1522	1 |- ((A i^i B) = (/) -> ((A X. C) i^i (B X. D)) = (/))

Syntax hints:   -> wi 3   = wceq 958   i^i cin 2049  (/)c0 2283   X. cxp 3174

This theorem is referenced by:  resdisj 3477  infxpidmlem11 7563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-rel 3191

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