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Theorem xpcan 5296
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 5295 . . 3  |-  ( ( C  =/=  (/)  /\  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  ( C  =  C  /\  A  =  B ) ) )
2 eqid 2435 . . . 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 2602 . . . 4  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
6 simpr 448 . . . . 5  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  A  =  (/) )
7 xpeq2 4884 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( C  X.  A )  =  ( C  X.  (/) ) )
8 xp0 5282 . . . . . . . . . 10  |-  ( C  X.  (/) )  =  (/)
97, 8syl6eq 2483 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( C  X.  A )  =  (/) )
109eqeq1d 2443 . . . . . . . 8  |-  ( A  =  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B )  <->  (/)  =  ( C  X.  B ) ) )
11 eqcom 2437 . . . . . . . 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 2600 . . . . . . . 8  |-  ( C  =/=  (/)  <->  -.  C  =  (/) )
15 xpeq0 5284 . . . . . . . . 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 2454 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  B )
226, 20, 21ee12an 1372 . . . 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 4884 . . 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 1652    =/= wne 2598   (/)c0 3620    X. cxp 4867
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-dm 4879  df-rn 4880
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