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Theorem xpdisj2 5034
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
StepHypRef Expression
1 inxp 4743 . 2  |-  ( ( C  X.  A )  i^i  ( D  X.  B ) )  =  ( ( C  i^i  D )  X.  ( A  i^i  B ) )
2 xpeq2 4624 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  i^i  D )  X.  ( A  i^i  B ) )  =  ( ( C  i^i  D
)  X.  (/) ) )
3 xp0 5028 . . 3  |-  ( ( C  i^i  D )  X.  (/) )  =  (/)
42, 3eqtrdi 2219 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  i^i  D )  X.  ( A  i^i  B ) )  =  (/) )
51, 4eqtrid 2215 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  X.  A )  i^i  ( D  X.  B ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    i^i cin 3120   (/)c0 3414    X. cxp 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617
This theorem is referenced by:  xpsndisj  5035
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