Theorem xpdisj2 5153

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 4855	. 2 |- ( ( C X. A ) i^i ( D X. B ) ) = ( ( C i^i D ) X. ( A i^i B ) )
2		xpeq2 4733	. . 3 |- ( ( A i^i B ) = (/) -> ( ( C i^i D ) X. ( A i^i B ) ) = ( ( C i^i D ) X. (/) ) )
3		xp0 5147	. . 3 |- ( ( C i^i D ) X. (/) ) = (/)
4	2, 3	eqtrdi 2278	. 2 |- ( ( A i^i B ) = (/) -> ( ( C i^i D ) X. ( A i^i B ) ) = (/) )
5	1, 4	eqtrid 2274	1 |- ( ( A i^i B ) = (/) -> ( ( C X. A ) i^i ( D X. B ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1395   i^i cin 3196   (/)c0 3491   X. cxp 4716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726

This theorem is referenced by:  xpsndisj  5154