Theorem isdrngo3 36044

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 36043	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7		eldifi 4057	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
8		difss 4062	. . . . . . . 8 ⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋
9		ssrexv 3984	. . . . . . . 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 3005	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍 ↔ 𝑈 ≠ 𝑍))
12	11	biimparc 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
13	3, 4, 1, 2	rngolz 36007	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑍𝐻𝑥) = 𝑍)
14		oveq1 7262	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
15	14	eqeq1d 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
16	13, 15	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
17	16	necon3d 2963	. . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ (𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦 ≠ 𝑍)
20	19	an4s 656	. . . . . . . . . . . . 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 4717	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍))
25	23, 24	syl6ibr 251	. . . . . . . . . 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 295	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈)))
30	29	reximdv2 3198	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
31	10, 30	impbid2 225	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
32	7, 31	sylan2 592	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
33	32	ralbidva 3119	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
34	33	pm5.32da 578	. . 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 274	1 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ⊆ wss 3883  {csn 4558  ran crn 5581  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  GIdcgi 28753  RingOpscrngo 35979  DivRingOpscdrng 36033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-grpo 28756  df-gid 28757  df-ginv 28758  df-ablo 28808  df-ass 35928  df-exid 35930  df-mgmOLD 35934  df-sgrOLD 35946  df-mndo 35952  df-rngo 35980  df-drngo 36034

This theorem is referenced by:  isfldidl  36153