Theorem isdrngo3 37925

Description: A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
isdivrng1.1	⊢ 𝐺 = (1st ‘𝑅)
isdivrng1.2	⊢ 𝐻 = (2nd ‘𝑅)
isdivrng1.3	⊢ 𝑍 = (GId‘𝐺)
isdivrng1.4	⊢ 𝑋 = ran 𝐺
isdivrng2.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
isdrngo3	⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))

Distinct variable groups:   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑍,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦

Allowed substitution hints:   𝐺(𝑥,𝑦)

Proof of Theorem isdrngo3

Step	Hyp	Ref	Expression
1		isdivrng1.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		isdivrng1.2	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
3		isdivrng1.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
4		isdivrng1.4	. . 3 ⊢ 𝑋 = ran 𝐺
5		isdivrng2.5	. . 3 ⊢ 𝑈 = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdrngo2 37924	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7		eldifi 4111	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
8		difss 4116	. . . . . . . 8 ⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋
9		ssrexv 4033	. . . . . . . 8 ⊢ ((𝑋 ∖ {𝑍}) ⊆ 𝑋 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
11		neeq1 2993	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍 ↔ 𝑈 ≠ 𝑍))
12	11	biimparc 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
13	3, 4, 1, 2	rngolz 37888	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑍𝐻𝑥) = 𝑍)
14		oveq1 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
15	14	eqeq1d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
16	13, 15	syl5ibrcom 247	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
17	16	necon3d 2952	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ((𝑦𝐻𝑥) ≠ 𝑍 → 𝑦 ≠ 𝑍))
18	17	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) ≠ 𝑍) → 𝑦 ≠ 𝑍)
19	12, 18	sylan2 593	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ (𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦 ≠ 𝑍)
20	19	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦 ≠ 𝑍)
21	20	anassrs 467	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → 𝑦 ≠ 𝑍)
22		pm3.2 469	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → (𝑦 ≠ 𝑍 → (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍)))
23	21, 22	syl5com 31	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ 𝑋 → (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍)))
24		eldifsn 4766	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍))
25	23, 24	imbitrrdi 252	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ 𝑋 → 𝑦 ∈ (𝑋 ∖ {𝑍})))
26	25	imdistanda 571	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))))
27		ancom 460	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ 𝑋))
28		ancom 460	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})))
29	26, 27, 28	3imtr4g 296	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈)))
30	29	reximdv2 3151	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
31	10, 30	impbid2 226	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
32	7, 31	sylan2 593	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
33	32	ralbidva 3163	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
34	33	pm5.32da 579	. . 3 ⊢ (𝑅 ∈ RingOps → ((𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))
35	34	pm5.32i 574	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))
36	6, 35	bitri 275	1 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∃wrex 3059   ∖ cdif 3928   ⊆ wss 3931  {csn 4606  ran crn 5666  ‘cfv 6541  (class class class)co 7413  1^st c1st 7994  2^nd c2nd 7995  GIdcgi 30437  RingOpscrngo 37860  DivRingOpscdrng 37914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-1st 7996  df-2nd 7997  df-1o 8488  df-en 8968  df-grpo 30440  df-gid 30441  df-ginv 30442  df-ablo 30492  df-ass 37809  df-exid 37811  df-mgmOLD 37815  df-sgrOLD 37827  df-mndo 37833  df-rngo 37861  df-drngo 37915

This theorem is referenced by:  isfldidl  38034