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Theorem xpcan 5474
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Proof of Theorem xpcan
StepHypRef Expression
1 xp11 5473 . . 3 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶𝐴 = 𝐵)))
2 eqid 2609 . . . 4 𝐶 = 𝐶
32biantrur 525 . . 3 (𝐴 = 𝐵 ↔ (𝐶 = 𝐶𝐴 = 𝐵))
41, 3syl6bbr 276 . 2 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5 nne 2785 . . . 4 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpr 475 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7 xpeq2 5042 . . . . . . . . . 10 (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8 xp0 5456 . . . . . . . . . 10 (𝐶 × ∅) = ∅
97, 8syl6eq 2659 . . . . . . . . 9 (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
109eqeq1d 2611 . . . . . . . 8 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11 eqcom 2616 . . . . . . . 8 (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
1210, 11syl6bb 274 . . . . . . 7 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
1312adantl 480 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14 df-ne 2781 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 5458 . . . . . . . . 9 ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16 orel1 395 . . . . . . . . 9 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
1715, 16syl5bi 230 . . . . . . . 8 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1814, 17sylbi 205 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1918adantr 479 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
2013, 19sylbid 228 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21 eqtr3 2630 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 565 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
235, 22sylan2b 490 . . 3 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24 xpeq2 5042 . . 3 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
2523, 24impbid1 213 . 2 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
264, 25pm2.61dan 827 1 (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wne 2779  c0 3873   × cxp 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038
This theorem is referenced by: (None)
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