Theorem xpcan2 6182

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

Step	Hyp	Ref	Expression
1		xp11 6180	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐶)))
2		eqid 2763	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantru 539	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐶))
4	1, 3	bitr4di 292	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5		nne 2963	. . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpl 488	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7		xpeq1 5670	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8		0xp 5756	. . . . . . . . . 10 ⊢ (∅ × 𝐶) = ∅
9	7, 8	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
10	9	eqeq1d 2765	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11		eqcom 2770	. . . . . . . 8 ⊢ (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
12	10, 11	bitrdi 290	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
13	12	adantr 486	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14		df-ne 2960	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 6164	. . . . . . . . 9 ⊢ ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16		orel2 904	. . . . . . . . 9 ⊢ (¬ 𝐶 = ∅ → ((𝐵 = ∅ ∨ 𝐶 = ∅) → 𝐵 = ∅))
17	15, 16	biimtrid 245	. . . . . . . 8 ⊢ (¬ 𝐶 = ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
18	14, 17	sylbi 220	. . . . . . 7 ⊢ (𝐶 ≠ ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
19	18	adantl 487	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
20	13, 19	sylbid 243	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21		eqtr3 2785	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 697	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23		xpeq1 5670	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
24	22, 23	impbid1 228	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
25	5, 24	sylanb 593	. 2 ⊢ ((¬ 𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
26	4, 25	pm2.61ian 824	1 ⊢ (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ≠ wne 2959  ∅c0 4281   × cxp 5654

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667

This theorem is used by: (None)