Theorem xpdisj2 4890

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

Assertion

Ref	Expression
xpdisj2	|- ( ( A i^i B ) = (/) -> ( ( C X. A ) i^i ( D X. B ) ) = (/) )

Proof of Theorem xpdisj2

Step	Hyp	Ref	Expression
1		inxp 4601	. 2 |- ( ( C X. A ) i^i ( D X. B ) ) = ( ( C i^i D ) X. ( A i^i B ) )
2		xpeq2 4482	. . 3 |- ( ( A i^i B ) = (/) -> ( ( C i^i D ) X. ( A i^i B ) ) = ( ( C i^i D ) X. (/) ) )
3		xp0 4884	. . 3 |- ( ( C i^i D ) X. (/) ) = (/)
4	2, 3	syl6eq 2143	. 2 |- ( ( A i^i B ) = (/) -> ( ( C i^i D ) X. ( A i^i B ) ) = (/) )
5	1, 4	syl5eq 2139	1 |- ( ( A i^i B ) = (/) -> ( ( C X. A ) i^i ( D X. B ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1296   i^i cin 3012   (/)c0 3302   X. cxp 4465

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-xp 4473  df-rel 4474  df-cnv 4475

This theorem is referenced by:  xpsndisj  4891