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Theorem xpcan2 5359
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan2  |-  ( C  =/=  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C
)  <->  A  =  B
) )

Proof of Theorem xpcan2
StepHypRef Expression
1 xp11 5357 . . 3  |-  ( ( A  =/=  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  ( A  =  B  /\  C  =  C ) ) )
2 eqid 2450 . . . 4  |-  C  =  C
32biantru 505 . . 3  |-  ( A  =  B  <->  ( A  =  B  /\  C  =  C ) )
41, 3syl6bbr 263 . 2  |-  ( ( A  =/=  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  A  =  B ) )
5 nne 2647 . . 3  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
6 simpl 457 . . . . 5  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  A  =  (/) )
7 xpeq1 4938 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( A  X.  C )  =  ( (/)  X.  C
) )
8 0xp 5001 . . . . . . . . . 10  |-  ( (/)  X.  C )  =  (/)
97, 8syl6eq 2506 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( A  X.  C )  =  (/) )
109eqeq1d 2452 . . . . . . . 8  |-  ( A  =  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C )  <->  (/)  =  ( B  X.  C ) ) )
11 eqcom 2458 . . . . . . . 8  |-  ( (/)  =  ( B  X.  C )  <->  ( B  X.  C )  =  (/) )
1210, 11syl6bb 261 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C )  <->  ( B  X.  C )  =  (/) ) )
1312adantr 465 . . . . . 6  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  ( B  X.  C )  =  (/) ) )
14 df-ne 2643 . . . . . . . 8  |-  ( C  =/=  (/)  <->  -.  C  =  (/) )
15 xpeq0 5342 . . . . . . . . 9  |-  ( ( B  X.  C )  =  (/)  <->  ( B  =  (/)  \/  C  =  (/) ) )
16 orel2 383 . . . . . . . . 9  |-  ( -.  C  =  (/)  ->  (
( B  =  (/)  \/  C  =  (/) )  ->  B  =  (/) ) )
1715, 16syl5bi 217 . . . . . . . 8  |-  ( -.  C  =  (/)  ->  (
( B  X.  C
)  =  (/)  ->  B  =  (/) ) )
1814, 17sylbi 195 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( ( B  X.  C )  =  (/)  ->  B  =  (/) ) )
1918adantl 466 . . . . . 6  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( B  X.  C
)  =  (/)  ->  B  =  (/) ) )
2013, 19sylbid 215 . . . . 5  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  ->  B  =  (/) ) )
21 eqtr3 2477 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  B )
226, 20, 21syl6an 545 . . . 4  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  ->  A  =  B )
)
23 xpeq1 4938 . . . 4  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
2422, 23impbid1 203 . . 3  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  A  =  B ) )
255, 24sylanb 472 . 2  |-  ( ( -.  A  =/=  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  A  =  B ) )
264, 25pm2.61ian 788 1  |-  ( C  =/=  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C
)  <->  A  =  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    =/= wne 2641   (/)c0 3721    X. cxp 4922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-xp 4930  df-rel 4931  df-cnv 4932  df-dm 4934  df-rn 4935
This theorem is referenced by:  vcoprnelem  24077
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