Theorem xpcan 6175

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 6174	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵)))
2		eqid 2732	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantrur 531	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐶 = 𝐶 ∧ 𝐴 = 𝐵))
4	1, 3	bitr4di 288	. 2 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5		nne 2944	. . . 4 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpr 485	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		xpeq2 5697	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8		xp0 6157	. . . . . . . . . 10 ⊢ (𝐶 × ∅) = ∅
9	7, 8	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
10	9	eqeq1d 2734	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11		eqcom 2739	. . . . . . . 8 ⊢ (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
12	10, 11	bitrdi 286	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
13	12	adantl 482	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14		df-ne 2941	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 6159	. . . . . . . . 9 ⊢ ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16		orel1 887	. . . . . . . . 9 ⊢ (¬ 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
17	15, 16	biimtrid 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 2758	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 682	. . . 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 5697	. . 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 811	1 ⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ≠ wne 2940  ∅c0 4322   × cxp 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687

This theorem is referenced by: (None)