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Theorem xpcan 5057
Description: Cancellation law for cross-product. (Contributed by set.mm contributors, 30-Aug-2011.)
Assertion
Ref Expression
xpcan (C → ((C × A) = (C × B) ↔ A = B))

Proof of Theorem xpcan
StepHypRef Expression
1 xp11 5056 . . 3 ((C A) → ((C × A) = (C × B) ↔ (C = C A = B)))
2 eqid 2353 . . . 4 C = C
32biantrur 492 . . 3 (A = B ↔ (C = C A = B))
41, 3syl6bbr 254 . 2 ((C A) → ((C × A) = (C × B) ↔ A = B))
5 nne 2520 . . . 4 AA = )
6 xpeq2 4799 . . . . . . . . . . 11 (A = → (C × A) = (C × ))
7 xp0 5044 . . . . . . . . . . 11 (C × ) =
86, 7syl6eq 2401 . . . . . . . . . 10 (A = → (C × A) = )
98eqeq1d 2361 . . . . . . . . 9 (A = → ((C × A) = (C × B) ↔ = (C × B)))
10 eqcom 2355 . . . . . . . . 9 ( = (C × B) ↔ (C × B) = )
119, 10syl6bb 252 . . . . . . . 8 (A = → ((C × A) = (C × B) ↔ (C × B) = ))
1211adantl 452 . . . . . . 7 ((C A = ) → ((C × A) = (C × B) ↔ (C × B) = ))
13 df-ne 2518 . . . . . . . . 9 (C ↔ ¬ C = )
14 xpeq0 5046 . . . . . . . . . 10 ((C × B) = ↔ (C = B = ))
15 orel1 371 . . . . . . . . . 10 C = → ((C = B = ) → B = ))
1614, 15syl5bi 208 . . . . . . . . 9 C = → ((C × B) = B = ))
1713, 16sylbi 187 . . . . . . . 8 (C → ((C × B) = B = ))
1817adantr 451 . . . . . . 7 ((C A = ) → ((C × B) = B = ))
1912, 18sylbid 206 . . . . . 6 ((C A = ) → ((C × A) = (C × B) → B = ))
20 simpr 447 . . . . . 6 ((C A = ) → A = )
2119, 20jctild 527 . . . . 5 ((C A = ) → ((C × A) = (C × B) → (A = B = )))
22 eqtr3 2372 . . . . 5 ((A = B = ) → A = B)
2321, 22syl6 29 . . . 4 ((C A = ) → ((C × A) = (C × B) → A = B))
245, 23sylan2b 461 . . 3 ((C ¬ A) → ((C × A) = (C × B) → A = B))
25 xpeq2 4799 . . 3 (A = B → (C × A) = (C × B))
2624, 25impbid1 194 . 2 ((C ¬ A) → ((C × A) = (C × B) ↔ A = B))
274, 26pm2.61dan 766 1 (C → ((C × A) = (C × B) ↔ A = B))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wo 357   wa 358   = wceq 1642  wne 2516  c0 3550   × cxp 4770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787
This theorem is referenced by: (None)
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