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Theorem difxp1 6154
Description: Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
difxp1  |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )

Proof of Theorem difxp1
StepHypRef Expression
1 difxp 6153 . 2  |-  ( ( A  X.  C ) 
\  ( B  X.  C ) )  =  ( ( ( A 
\  B )  X.  C )  u.  ( A  X.  ( C  \  C ) ) )
2 difid 3522 . . . . 5  |-  ( C 
\  C )  =  (/)
32xpeq2i 4710 . . . 4  |-  ( A  X.  ( C  \  C ) )  =  ( A  X.  (/) )
4 xp0 5098 . . . 4  |-  ( A  X.  (/) )  =  (/)
53, 4eqtri 2303 . . 3  |-  ( A  X.  ( C  \  C ) )  =  (/)
65uneq2i 3326 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  ( A  X.  ( C  \  C ) ) )  =  ( ( ( A  \  B )  X.  C
)  u.  (/) )
7 un0 3479 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  (/) )  =  ( ( A  \  B
)  X.  C )
81, 6, 73eqtrri 2308 1  |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    \ cdif 3149    u. cun 3150   (/)c0 3455    X. cxp 4687
This theorem is referenced by:  dfsup3OLD  7197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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