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Theorem xpcan2 5058

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 5056	. . 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 2520	. . 3 ⊢ (¬ A ≠ ∅ ↔ A = ∅)
6		xpeq1 4798	. . . . . . . . . . 11 ⊢ (A = ∅ → (A × C) = (∅ × C))
7		xp0r 4842	. . . . . . . . . . 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 2518	. . . . . . . . 9 ⊢ (C ≠ ∅ ↔ ¬ C = ∅)
14		xpeq0 5046	. . . . . . . . . 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 4798	. . . 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))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∨ wo 357 ∧ wa 358 = wceq 1642 ≠ wne 2516 ∅c0 3550 × cxp 4770

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787

This theorem is referenced by: (None)

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