Theorem xpcan 5489

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 5488	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵)))
2		eqid 2610	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantrur 526	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵))
4	1, 3	syl6bbr 277	. 2 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5		nne 2786	. . . 4 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpr 476	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		xpeq2 5053	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8		xp0 5471	. . . . . . . . . 10 ⊢ (𝐶 × ∅) = ∅
9	7, 8	syl6eq 2660	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
10	9	eqeq1d 2612	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11		eqcom 2617	. . . . . . . 8 ⊢ (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
12	10, 11	syl6bb 275	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
13	12	adantl 481	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14		df-ne 2782	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 5473	. . . . . . . . 9 ⊢ ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16		orel1 396	. . . . . . . . 9 ⊢ (¬ 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
17	15, 16	syl5bi 231	. . . . . . . 8 ⊢ (¬ 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
18	14, 17	sylbi 206	. . . . . . 7 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
19	18	adantr 480	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
20	13, 19	sylbid 229	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21		eqtr3 2631	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 566	. . . 4 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
23	5, 22	sylan2b 491	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24		xpeq2 5053	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
25	23, 24	impbid1 214	. 2 ⊢ ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
26	4, 25	pm2.61dan 828	1 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ≠ wne 2780  ∅c0 3874   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049

This theorem is referenced by: (None)