Theorem xpcan 6079

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 6078	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵)))
2		eqid 2738	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantrur 531	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵))
4	1, 3	bitr4di 289	. 2 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5		nne 2947	. . . 4 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpr 485	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		xpeq2 5610	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8		xp0 6061	. . . . . . . . . 10 ⊢ (𝐶 × ∅) = ∅
9	7, 8	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
10	9	eqeq1d 2740	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11		eqcom 2745	. . . . . . . 8 ⊢ (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
12	10, 11	bitrdi 287	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
13	12	adantl 482	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14		df-ne 2944	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 6063	. . . . . . . . 9 ⊢ ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16		orel1 886	. . . . . . . . 9 ⊢ (¬ 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
17	15, 16	syl5bi 241	. . . . . . . 8 ⊢ (¬ 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
18	14, 17	sylbi 216	. . . . . . 7 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
19	18	adantr 481	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
20	13, 19	sylbid 239	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21		eqtr3 2764	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 681	. . . 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 5610	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
25	23, 24	impbid1 224	. 2 ⊢ ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
26	4, 25	pm2.61dan 810	1 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ≠ wne 2943  ∅c0 4256   × cxp 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600

This theorem is referenced by: (None)