Theorem xpcan2 5059

Description: Cancellation law for cross-product. (Contributed by set.mm contributors, 30-Aug-2011.)

Assertion

Ref	Expression
xpcan2	⊢ (C ≠ ∅ → ((A × C) = (B × C) ↔ A = B))

Proof of Theorem xpcan2

Step	Hyp	Ref	Expression
1		xp11 5057	. . 3 ⊢ ((A ≠ ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) ↔ (A = B ∧ C = C)))
2		eqid 2353	. . . 4 ⊢ C = C
3	2	biantru 491	. . 3 ⊢ (A = B ↔ (A = B ∧ C = C))
4	1, 3	syl6bbr 254	. 2 ⊢ ((A ≠ ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) ↔ A = B))
5		nne 2521	. . 3 ⊢ (¬ A ≠ ∅ ↔ A = ∅)
6		xpeq1 4799	. . . . . . . . . . 11 ⊢ (A = ∅ → (A × C) = (∅ × C))
7		xp0r 4843	. . . . . . . . . . 11 ⊢ (∅ × C) = ∅
8	6, 7	syl6eq 2401	. . . . . . . . . 10 ⊢ (A = ∅ → (A × C) = ∅)
9	8	eqeq1d 2361	. . . . . . . . 9 ⊢ (A = ∅ → ((A × C) = (B × C) ↔ ∅ = (B × C)))
10		eqcom 2355	. . . . . . . . 9 ⊢ (∅ = (B × C) ↔ (B × C) = ∅)
11	9, 10	syl6bb 252	. . . . . . . 8 ⊢ (A = ∅ → ((A × C) = (B × C) ↔ (B × C) = ∅))
12	11	adantr 451	. . . . . . 7 ⊢ ((A = ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) ↔ (B × C) = ∅))
13		df-ne 2519	. . . . . . . . 9 ⊢ (C ≠ ∅ ↔ ¬ C = ∅)
14		xpeq0 5047	. . . . . . . . . 10 ⊢ ((B × C) = ∅ ↔ (B = ∅ ∨ C = ∅))
15		orel2 372	. . . . . . . . . 10 ⊢ (¬ C = ∅ → ((B = ∅ ∨ C = ∅) → B = ∅))
16	14, 15	syl5bi 208	. . . . . . . . 9 ⊢ (¬ C = ∅ → ((B × C) = ∅ → B = ∅))
17	13, 16	sylbi 187	. . . . . . . 8 ⊢ (C ≠ ∅ → ((B × C) = ∅ → B = ∅))
18	17	adantl 452	. . . . . . 7 ⊢ ((A = ∅ ∧ C ≠ ∅) → ((B × C) = ∅ → B = ∅))
19	12, 18	sylbid 206	. . . . . 6 ⊢ ((A = ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) → B = ∅))
20		simpl 443	. . . . . 6 ⊢ ((A = ∅ ∧ C ≠ ∅) → A = ∅)
21	19, 20	jctild 527	. . . . 5 ⊢ ((A = ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) → (A = ∅ ∧ B = ∅)))
22		eqtr3 2372	. . . . 5 ⊢ ((A = ∅ ∧ B = ∅) → A = B)
23	21, 22	syl6 29	. . . 4 ⊢ ((A = ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) → A = B))
24		xpeq1 4799	. . . 4 ⊢ (A = B → (A × C) = (B × C))
25	23, 24	impbid1 194	. . 3 ⊢ ((A = ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) ↔ A = B))
26	5, 25	sylanb 458	. 2 ⊢ ((¬ A ≠ ∅ ∧ C ≠ ∅) → ((A × C) = (B × C) ↔ A = B))
27	4, 26	pm2.61ian 765	1 ⊢ (C ≠ ∅ → ((A × C) = (B × C) ↔ A = B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ≠ wne 2517  ∅c0 3551   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by: (None)