New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  xpcan2 GIF version

Theorem xpcan2 5058
 Description: Cancellation law for cross-product. (Contributed by set.mm contributors, 30-Aug-2011.)
Assertion
Ref Expression
xpcan2 (C → ((A × C) = (B × C) ↔ A = B))

Proof of Theorem xpcan2
StepHypRef Expression
1 xp11 5056 . . 3 ((A C) → ((A × C) = (B × C) ↔ (A = B C = C)))
2 eqid 2353 . . . 4 C = C
32biantru 491 . . 3 (A = B ↔ (A = B C = C))
41, 3syl6bbr 254 . 2 ((A C) → ((A × C) = (B × C) ↔ A = B))
5 nne 2520 . . 3 AA = )
6 xpeq1 4798 . . . . . . . . . . 11 (A = → (A × C) = ( × C))
7 xp0r 4842 . . . . . . . . . . 11 ( × C) =
86, 7syl6eq 2401 . . . . . . . . . 10 (A = → (A × C) = )
98eqeq1d 2361 . . . . . . . . 9 (A = → ((A × C) = (B × C) ↔ = (B × C)))
10 eqcom 2355 . . . . . . . . 9 ( = (B × C) ↔ (B × C) = )
119, 10syl6bb 252 . . . . . . . 8 (A = → ((A × C) = (B × C) ↔ (B × C) = ))
1211adantr 451 . . . . . . 7 ((A = C) → ((A × C) = (B × C) ↔ (B × C) = ))
13 df-ne 2518 . . . . . . . . 9 (C ↔ ¬ C = )
14 xpeq0 5046 . . . . . . . . . 10 ((B × C) = ↔ (B = C = ))
15 orel2 372 . . . . . . . . . 10 C = → ((B = C = ) → B = ))
1614, 15syl5bi 208 . . . . . . . . 9 C = → ((B × C) = B = ))
1713, 16sylbi 187 . . . . . . . 8 (C → ((B × C) = B = ))
1817adantl 452 . . . . . . 7 ((A = C) → ((B × C) = B = ))
1912, 18sylbid 206 . . . . . 6 ((A = C) → ((A × C) = (B × C) → B = ))
20 simpl 443 . . . . . 6 ((A = C) → A = )
2119, 20jctild 527 . . . . 5 ((A = C) → ((A × C) = (B × C) → (A = B = )))
22 eqtr3 2372 . . . . 5 ((A = B = ) → A = B)
2321, 22syl6 29 . . . 4 ((A = C) → ((A × C) = (B × C) → A = B))
24 xpeq1 4798 . . . 4 (A = B → (A × C) = (B × C))
2523, 24impbid1 194 . . 3 ((A = C) → ((A × C) = (B × C) ↔ A = B))
265, 25sylanb 458 . 2 ((¬ A C) → ((A × C) = (B × C) ↔ A = B))
274, 26pm2.61ian 765 1 (C → ((A × C) = (B × C) ↔ 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)
 Copyright terms: Public domain W3C validator