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Theorem difxp1 6382
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 6381 . 2  |-  ( ( A  X.  C ) 
\  ( B  X.  C ) )  =  ( ( ( A 
\  B )  X.  C )  u.  ( A  X.  ( C  \  C ) ) )
2 difid 3697 . . . . 5  |-  ( C 
\  C )  =  (/)
32xpeq2i 4900 . . . 4  |-  ( A  X.  ( C  \  C ) )  =  ( A  X.  (/) )
4 xp0 5292 . . . 4  |-  ( A  X.  (/) )  =  (/)
53, 4eqtri 2457 . . 3  |-  ( A  X.  ( C  \  C ) )  =  (/)
65uneq2i 3499 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  ( A  X.  ( C  \  C ) ) )  =  ( ( ( A  \  B )  X.  C
)  u.  (/) )
7 un0 3653 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  (/) )  =  ( ( A  \  B
)  X.  C )
81, 6, 73eqtrri 2462 1  |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    \ cdif 3318    u. cun 3319   (/)c0 3629    X. cxp 4877
This theorem is referenced by:  dfsup3OLD  7450  sxbrsigalem2  24637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887
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