Theorem xpcan 6176

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 6175	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵)))
2		eqid 2765	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantrur 540	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵))
4	1, 3	bitr4di 292	. 2 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5		nne 2964	. . . 4 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpr 490	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		xpeq2 5684	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8		xp0 5763	. . . . . . . . . 10 ⊢ (𝐶 × ∅) = ∅
9	7, 8	eqtrdi 2816	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
10	9	eqeq1d 2767	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11		eqcom 2772	. . . . . . . 8 ⊢ (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
12	10, 11	bitrdi 290	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
13	12	adantl 487	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14		df-ne 2961	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 6159	. . . . . . . . 9 ⊢ ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16		orel1 902	. . . . . . . . 9 ⊢ (¬ 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
17	15, 16	biimtrid 245	. . . . . . . 8 ⊢ (¬ 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
18	14, 17	sylbi 220	. . . . . . 7 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
19	18	adantr 486	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
20	13, 19	sylbid 243	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21		eqtr3 2787	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 697	. . . 4 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
23	5, 22	sylan2b 606	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24		xpeq2 5684	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
25	23, 24	impbid1 228	. 2 ⊢ ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
26	4, 25	pm2.61dan 825	1 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ≠ wne 2960  ∅c0 4286   × cxp 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  diag1f1lem  50117  diag2f1lem  50119