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Theorem difxp2 6382
Description: Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
difxp2  |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )

Proof of Theorem difxp2
StepHypRef Expression
1 difxp 6380 . 2  |-  ( ( A  X.  B ) 
\  ( A  X.  C ) )  =  ( ( ( A 
\  A )  X.  B )  u.  ( A  X.  ( B  \  C ) ) )
2 difid 3696 . . . . 5  |-  ( A 
\  A )  =  (/)
32xpeq1i 4898 . . . 4  |-  ( ( A  \  A )  X.  B )  =  ( (/)  X.  B
)
4 xp0r 4956 . . . 4  |-  ( (/)  X.  B )  =  (/)
53, 4eqtri 2456 . . 3  |-  ( ( A  \  A )  X.  B )  =  (/)
65uneq1i 3497 . 2  |-  ( ( ( A  \  A
)  X.  B )  u.  ( A  X.  ( B  \  C ) ) )  =  (
(/)  u.  ( A  X.  ( B  \  C
) ) )
7 uncom 3491 . . 3  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( ( A  X.  ( B 
\  C ) )  u.  (/) )
8 un0 3652 . . 3  |-  ( ( A  X.  ( B 
\  C ) )  u.  (/) )  =  ( A  X.  ( B 
\  C ) )
97, 8eqtri 2456 . 2  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( A  X.  ( B  \  C ) )
101, 6, 93eqtrri 2461 1  |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3317    u. cun 3318   (/)c0 3628    X. cxp 4876
This theorem is referenced by:  imadifxp  24038  sxbrsigalem2  24636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-rel 4885
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