Theorem xpcan2 6166

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 6164	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐶)))
2		eqid 2724	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantru 529	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐶))
4	1, 3	bitr4di 289	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5		nne 2936	. . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpl 482	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7		xpeq1 5680	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8		0xp 5764	. . . . . . . . . 10 ⊢ (∅ × 𝐶) = ∅
9	7, 8	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
10	9	eqeq1d 2726	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11		eqcom 2731	. . . . . . . 8 ⊢ (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
12	10, 11	bitrdi 287	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
13	12	adantr 480	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14		df-ne 2933	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 6149	. . . . . . . . 9 ⊢ ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16		orel2 887	. . . . . . . . 9 ⊢ (¬ 𝐶 = ∅ → ((𝐵 = ∅ ∨ 𝐶 = ∅) → 𝐵 = ∅))
17	15, 16	biimtrid 241	. . . . . . . 8 ⊢ (¬ 𝐶 = ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
18	14, 17	sylbi 216	. . . . . . 7 ⊢ (𝐶 ≠ ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
19	18	adantl 481	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
20	13, 19	sylbid 239	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21		eqtr3 2750	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 681	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23		xpeq1 5680	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
24	22, 23	impbid1 224	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
25	5, 24	sylanb 580	. 2 ⊢ ((¬ 𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
26	4, 25	pm2.61ian 809	1 ⊢ (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ≠ wne 2932  ∅c0 4314   × cxp 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677

This theorem is referenced by: (None)