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Theorem xpcan2 6183
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan2 (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Proof of Theorem xpcan2
StepHypRef Expression
1 xp11 6181 . . 3 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐴 = 𝐵𝐶 = 𝐶)))
2 eqid 2725 . . . 4 𝐶 = 𝐶
32biantru 528 . . 3 (𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐶 = 𝐶))
41, 3bitr4di 288 . 2 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5 nne 2933 . . 3 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpl 481 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7 xpeq1 5692 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8 0xp 5776 . . . . . . . . . 10 (∅ × 𝐶) = ∅
97, 8eqtrdi 2781 . . . . . . . . 9 (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
109eqeq1d 2727 . . . . . . . 8 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11 eqcom 2732 . . . . . . . 8 (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
1210, 11bitrdi 286 . . . . . . 7 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
1312adantr 479 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14 df-ne 2930 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 6166 . . . . . . . . 9 ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16 orel2 888 . . . . . . . . 9 𝐶 = ∅ → ((𝐵 = ∅ ∨ 𝐶 = ∅) → 𝐵 = ∅))
1715, 16biimtrid 241 . . . . . . . 8 𝐶 = ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1814, 17sylbi 216 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1918adantl 480 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
2013, 19sylbid 239 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21 eqtr3 2751 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 682 . . . 4 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23 xpeq1 5692 . . . 4 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2422, 23impbid1 224 . . 3 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
255, 24sylanb 579 . 2 ((¬ 𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
264, 25pm2.61ian 810 1 (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wne 2929  c0 4322   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689
This theorem is referenced by: (None)
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