Theorem xpcan2 6167

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 6165	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐶)))
2		eqid 2765	. . . 4 ⊢ 𝐶 = 𝐶
3	2	biantru 538	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐶))
4	1, 3	bitr4di 292	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5		nne 2964	. . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6		simpl 487	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7		xpeq1 5666	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8		0xp 5751	. . . . . . . . . 10 ⊢ (∅ × 𝐶) = ∅
9	7, 8	eqtrdi 2816	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
10	9	eqeq1d 2767	. . . . . . . 8 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11		eqcom 2772	. . . . . . . 8 ⊢ (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
12	10, 11	bitrdi 290	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
13	12	adantr 485	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14		df-ne 2961	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15		xpeq0 6149	. . . . . . . . 9 ⊢ ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16		orel2 903	. . . . . . . . 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 486	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
20	13, 19	sylbid 243	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21		eqtr3 2787	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
22	6, 20, 21	syl6an 696	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23		xpeq1 5666	. . . 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 592	. 2 ⊢ ((¬ 𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
26	4, 25	pm2.61ian 823	1 ⊢ (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ≠ wne 2960  ∅c0 4288   × cxp 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663

This theorem is referenced by: (None)