Theorem isdrngo3 37966

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 37965	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7		eldifi 4131	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
8		difss 4136	. . . . . . . 8 ⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋
9		ssrexv 4053	. . . . . . . 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 3003	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍 ↔ 𝑈 ≠ 𝑍))
12	11	biimparc 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
13	3, 4, 1, 2	rngolz 37929	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑍𝐻𝑥) = 𝑍)
14		oveq1 7438	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
15	14	eqeq1d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
16	13, 15	syl5ibrcom 247	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
17	16	necon3d 2961	. . . . . . . . . . . . . . . 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 4786	. . . . . . . . . . 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 3164	. . . . . . 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 3176	. . . 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 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ⊆ wss 3951  {csn 4626  ran crn 5686  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  GIdcgi 30509  RingOpscrngo 37901  DivRingOpscdrng 37955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-drngo 37956

This theorem is referenced by:  isfldidl  38075