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Theorem xpcan 5340
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 5339 . . 3  |-  ( ( C  =/=  (/)  /\  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  ( C  =  C  /\  A  =  B ) ) )
2 eqid 2443 . . . 4  |-  C  =  C
32biantrur 494 . . 3  |-  ( A  =  B  <->  ( C  =  C  /\  A  =  B ) )
41, 3syl6bbr 256 . 2  |-  ( ( C  =/=  (/)  /\  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  A  =  B ) )
5 nne 2612 . . . 4  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
6 simpr 449 . . . . 5  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  A  =  (/) )
7 xpeq2 4928 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( C  X.  A )  =  ( C  X.  (/) ) )
8 xp0 5326 . . . . . . . . . 10  |-  ( C  X.  (/) )  =  (/)
97, 8syl6eq 2491 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( C  X.  A )  =  (/) )
109eqeq1d 2451 . . . . . . . 8  |-  ( A  =  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B )  <->  (/)  =  ( C  X.  B ) ) )
11 eqcom 2445 . . . . . . . 8  |-  ( (/)  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) )
1210, 11syl6bb 254 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) ) )
1312adantl 454 . . . . . 6  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) ) )
14 df-ne 2608 . . . . . . . 8  |-  ( C  =/=  (/)  <->  -.  C  =  (/) )
15 xpeq0 5328 . . . . . . . . 9  |-  ( ( C  X.  B )  =  (/)  <->  ( C  =  (/)  \/  B  =  (/) ) )
16 orel1 373 . . . . . . . . 9  |-  ( -.  C  =  (/)  ->  (
( C  =  (/)  \/  B  =  (/) )  ->  B  =  (/) ) )
1715, 16syl5bi 210 . . . . . . . 8  |-  ( -.  C  =  (/)  ->  (
( C  X.  B
)  =  (/)  ->  B  =  (/) ) )
1814, 17sylbi 189 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( ( C  X.  B )  =  (/)  ->  B  =  (/) ) )
1918adantr 453 . . . . . 6  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  B
)  =  (/)  ->  B  =  (/) ) )
2013, 19sylbid 208 . . . . 5  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  B  =  (/) ) )
21 eqtr3 2462 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  B )
226, 20, 21ee12an 1373 . . . 4  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  A  =  B )
)
235, 22sylan2b 463 . . 3  |-  ( ( C  =/=  (/)  /\  -.  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  A  =  B )
)
24 xpeq2 4928 . . 3  |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B
) )
2523, 24impbid1 196 . 2  |-  ( ( C  =/=  (/)  /\  -.  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  A  =  B ) )
264, 25pm2.61dan 768 1  |-  ( C  =/=  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B
)  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1654    =/= wne 2606   (/)c0 3616    X. cxp 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924
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