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Theorem xpcan2 6199
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 6197 . . 3 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐴 = 𝐵𝐶 = 𝐶)))
2 eqid 2735 . . . 4 𝐶 = 𝐶
32biantru 529 . . 3 (𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐶 = 𝐶))
41, 3bitr4di 289 . 2 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5 nne 2942 . . 3 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpl 482 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7 xpeq1 5703 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8 0xp 5787 . . . . . . . . . 10 (∅ × 𝐶) = ∅
97, 8eqtrdi 2791 . . . . . . . . 9 (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
109eqeq1d 2737 . . . . . . . 8 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11 eqcom 2742 . . . . . . . 8 (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
1210, 11bitrdi 287 . . . . . . 7 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
1312adantr 480 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14 df-ne 2939 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 6182 . . . . . . . . 9 ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16 orel2 890 . . . . . . . . 9 𝐶 = ∅ → ((𝐵 = ∅ ∨ 𝐶 = ∅) → 𝐵 = ∅))
1715, 16biimtrid 242 . . . . . . . 8 𝐶 = ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1814, 17sylbi 217 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1918adantl 481 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
2013, 19sylbid 240 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21 eqtr3 2761 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 684 . . . 4 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23 xpeq1 5703 . . . 4 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2422, 23impbid1 225 . . 3 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
255, 24sylanb 581 . 2 ((¬ 𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
264, 25pm2.61ian 812 1 (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wne 2938  c0 4339   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by: (None)
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