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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 2819 . . . 4 𝐶 = 𝐶
32biantru 532 . . 3 (𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐶 = 𝐶))
41, 3syl6bbr 291 . 2 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5 nne 3018 . . 3 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpl 485 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7 xpeq1 5562 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8 0xp 5642 . . . . . . . . . 10 (∅ × 𝐶) = ∅
97, 8syl6eq 2870 . . . . . . . . 9 (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
109eqeq1d 2821 . . . . . . . 8 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11 eqcom 2826 . . . . . . . 8 (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
1210, 11syl6bb 289 . . . . . . 7 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
1312adantr 483 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14 df-ne 3015 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 6010 . . . . . . . . 9 ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16 orel2 887 . . . . . . . . 9 𝐶 = ∅ → ((𝐵 = ∅ ∨ 𝐶 = ∅) → 𝐵 = ∅))
1715, 16syl5bi 244 . . . . . . . 8 𝐶 = ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1814, 17sylbi 219 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1918adantl 484 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
2013, 19sylbid 242 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21 eqtr3 2841 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 682 . . . 4 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23 xpeq1 5562 . . . 4 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2422, 23impbid1 227 . . 3 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
255, 24sylanb 583 . 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 208  wa 398  wo 843   = wceq 1531  wne 3014  c0 4289   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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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