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Theorem xpcan2 6027
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 6025 . . 3 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐴 = 𝐵𝐶 = 𝐶)))
2 eqid 2818 . . . 4 𝐶 = 𝐶
32biantru 530 . . 3 (𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐶 = 𝐶))
41, 3syl6bbr 290 . 2 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5 nne 3017 . . 3 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpl 483 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7 xpeq1 5562 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8 0xp 5642 . . . . . . . . . 10 (∅ × 𝐶) = ∅
97, 8syl6eq 2869 . . . . . . . . 9 (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
109eqeq1d 2820 . . . . . . . 8 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11 eqcom 2825 . . . . . . . 8 (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
1210, 11syl6bb 288 . . . . . . 7 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
1312adantr 481 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14 df-ne 3014 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 6010 . . . . . . . . 9 ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16 orel2 884 . . . . . . . . 9 𝐶 = ∅ → ((𝐵 = ∅ ∨ 𝐶 = ∅) → 𝐵 = ∅))
1715, 16syl5bi 243 . . . . . . . 8 𝐶 = ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1814, 17sylbi 218 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1918adantl 482 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
2013, 19sylbid 241 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21 eqtr3 2840 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 680 . . . 4 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23 xpeq1 5562 . . . 4 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2422, 23impbid1 226 . . 3 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
255, 24sylanb 581 . 2 ((¬ 𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
264, 25pm2.61ian 808 1 (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wne 3013  c0 4288   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by: (None)
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