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Theorem xpcan 5246
Description: Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan  |-  ( C  =/=  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B
)  <->  A  =  B
) )

Proof of Theorem xpcan
StepHypRef Expression
1 xp11 5245 . . 3  |-  ( ( C  =/=  (/)  /\  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  ( C  =  C  /\  A  =  B ) ) )
2 eqid 2388 . . . 4  |-  C  =  C
32biantrur 493 . . 3  |-  ( A  =  B  <->  ( C  =  C  /\  A  =  B ) )
41, 3syl6bbr 255 . 2  |-  ( ( C  =/=  (/)  /\  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  A  =  B ) )
5 nne 2555 . . . 4  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
6 simpr 448 . . . . 5  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  A  =  (/) )
7 xpeq2 4834 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( C  X.  A )  =  ( C  X.  (/) ) )
8 xp0 5232 . . . . . . . . . 10  |-  ( C  X.  (/) )  =  (/)
97, 8syl6eq 2436 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( C  X.  A )  =  (/) )
109eqeq1d 2396 . . . . . . . 8  |-  ( A  =  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B )  <->  (/)  =  ( C  X.  B ) ) )
11 eqcom 2390 . . . . . . . 8  |-  ( (/)  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) )
1210, 11syl6bb 253 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) ) )
1312adantl 453 . . . . . 6  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) ) )
14 df-ne 2553 . . . . . . . 8  |-  ( C  =/=  (/)  <->  -.  C  =  (/) )
15 xpeq0 5234 . . . . . . . . 9  |-  ( ( C  X.  B )  =  (/)  <->  ( C  =  (/)  \/  B  =  (/) ) )
16 orel1 372 . . . . . . . . 9  |-  ( -.  C  =  (/)  ->  (
( C  =  (/)  \/  B  =  (/) )  ->  B  =  (/) ) )
1715, 16syl5bi 209 . . . . . . . 8  |-  ( -.  C  =  (/)  ->  (
( C  X.  B
)  =  (/)  ->  B  =  (/) ) )
1814, 17sylbi 188 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( ( C  X.  B )  =  (/)  ->  B  =  (/) ) )
1918adantr 452 . . . . . 6  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  B
)  =  (/)  ->  B  =  (/) ) )
2013, 19sylbid 207 . . . . 5  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  B  =  (/) ) )
21 eqtr3 2407 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  B )
226, 20, 21ee12an 1369 . . . 4  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  A  =  B )
)
235, 22sylan2b 462 . . 3  |-  ( ( C  =/=  (/)  /\  -.  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  A  =  B )
)
24 xpeq2 4834 . . 3  |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B
) )
2523, 24impbid1 195 . 2  |-  ( ( C  =/=  (/)  /\  -.  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  A  =  B ) )
264, 25pm2.61dan 767 1  |-  ( C  =/=  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B
)  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    =/= wne 2551   (/)c0 3572    X. cxp 4817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-dm 4829  df-rn 4830
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