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Theorem xpdisj1 4747
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
StepHypRef Expression
1 inxp 4470 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  D ) )
2 xpeq1 4359 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (
(/)  X.  ( C  i^i  D ) ) )
3 0xp 4420 . . 3  |-  ( (/)  X.  ( C  i^i  D
) )  =  (/)
42, 3syl6eq 2088 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (/) )
51, 4syl5eq 2084 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    i^i cin 2916   (/)c0 3224    X. cxp 4343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352
This theorem is referenced by:  djudisj  4750
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