Theorem isdrngo3 32928

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 32927	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7		eldifi 3694	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
8		difss 3699	. . . . . . . 8 ⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋
9		ssrexv 3630	. . . . . . . 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 2844	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍 ↔ 𝑈 ≠ 𝑍))
12	11	biimparc 503	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
13	3, 4, 1, 2	rngolz 32891	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑍𝐻𝑥) = 𝑍)
14		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
15	14	eqeq1d 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
16	13, 15	syl5ibrcom 236	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
17	16	necon3d 2803	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ((𝑦𝐻𝑥) ≠ 𝑍 → 𝑦 ≠ 𝑍))
18	17	imp 444	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) ≠ 𝑍) → 𝑦 ≠ 𝑍)
19	12, 18	sylan2 490	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ (𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦 ≠ 𝑍)
20	19	an4s 865	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦 ≠ 𝑍)
21	20	anassrs 678	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → 𝑦 ≠ 𝑍)
22		pm3.2 462	. . . . . . . . . . . 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 4260	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍))
25	23, 24	syl6ibr 241	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ 𝑋 → 𝑦 ∈ (𝑋 ∖ {𝑍})))
26	25	imdistanda 725	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))))
27		ancom 465	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ 𝑋))
28		ancom 465	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈 ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})))
29	26, 27, 28	3imtr4g 284	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈)))
30	29	reximdv2 2997	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
31	10, 30	impbid2 215	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
32	7, 31	sylan2 490	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
33	32	ralbidva 2968	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
34	33	pm5.32da 671	. . 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 667	. 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 263	1 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  GIdcgi 26728  RingOpscrngo 32863  DivRingOpscdrng 32917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-drngo 32918

This theorem is referenced by:  isfldidl  33037