Theorem xpdisj1 3468

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 3200	. . 3 |- ((A i^i B) = (/) -> ((A i^i B) X. (C i^i D)) = ((/) X. (C i^i D)))
2		xp0r 3239	. . 3 |- ((/) X. (C i^i D)) = (/)
3	1, 2	syl6eq 1523	. 2 |- ((A i^i B) = (/) -> ((A i^i B) X. (C i^i D)) = (/))
4		inxp 3269	. 2 |- ((A X. C) i^i (B X. D)) = ((A i^i B) X. (C i^i D))
5	3, 4	syl5eq 1519	1 |- ((A i^i B) = (/) -> ((A X. C) i^i (B X. D)) = (/))

Syntax hints:   -> wi 3   = wceq 956   i^i cin 2046  (/)c0 2280   X. cxp 3168

This theorem is referenced by:  resdisj 3471  infxpidmlem11 7562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185

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