Theorem xpcan 6149

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

Step	Hyp	Ref	Expression
1		xp11 6148	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵)))
2		eqid 2729	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantrur 530	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵))
4	1, 3	bitr4di 289	. 2 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5		nne 2929	. . . 4 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpr 484	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		xpeq2 5659	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8		xp0 6131	. . . . . . . . . 10 ⊢ (𝐶 × ∅) = ∅
9	7, 8	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
10	9	eqeq1d 2731	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11		eqcom 2736	. . . . . . . 8 ⊢ (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
12	10, 11	bitrdi 287	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
13	12	adantl 481	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14		df-ne 2926	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 6133	. . . . . . . . 9 ⊢ ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16		orel1 888	. . . . . . . . 9 ⊢ (¬ 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
17	15, 16	biimtrid 242	. . . . . . . 8 ⊢ (¬ 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
18	14, 17	sylbi 217	. . . . . . 7 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
19	18	adantr 480	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
20	13, 19	sylbid 240	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21		eqtr3 2751	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 684	. . . 4 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
23	5, 22	sylan2b 594	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24		xpeq2 5659	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
25	23, 24	impbid1 225	. 2 ⊢ ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
26	4, 25	pm2.61dan 812	1 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ≠ wne 2925  ∅c0 4296   × cxp 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649

This theorem is referenced by:  diag1f1lem  49295  diag2f1lem  49297