Theorem isdrngo3 38638

Description: Obsolete theorem, use isdrng5 20863 instead. A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 38637	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7		eldifi 4085	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
8		difss 4090	. . . . . . . 8 ⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋
9		ssrexv 4007	. . . . . . . 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 3020	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍 ↔ 𝑈 ≠ 𝑍))
12	11	biimparc 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
13	3, 4, 1, 2	rngolz 38601	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑍𝐻𝑥) = 𝑍)
14		oveq1 7417	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
15	14	eqeq1d 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
16	13, 15	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
17	16	necon3d 2979	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ((𝑦𝐻𝑥) ≠ 𝑍 → 𝑦 ≠ 𝑍))
18	17	imp 411	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) ≠ 𝑍) → 𝑦 ≠ 𝑍)
19	12, 18	sylan2 604	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ (𝑈 ≠ 𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦 ≠ 𝑍)
20	19	an4s 672	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦 ≠ 𝑍)
21	20	anassrs 472	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → 𝑦 ≠ 𝑍)
22		pm3.2 474	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → (𝑦 ≠ 𝑍 → (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍)))
23	21, 22	syl5com 32	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ 𝑋 → (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍)))
24		eldifsn 4753	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝑍))
25	23, 24	imbitrrdi 255	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ 𝑋 → 𝑦 ∈ (𝑋 ∖ {𝑍})))
26	25	imdistanda 581	. . . . . . . . 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 299	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈)))
30	29	reximdv2 3175	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
31	10, 30	impbid2 229	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
32	7, 31	sylan2 604	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
33	32	ralbidva 3186	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))
34	33	pm5.32da 589	. . 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 584	. 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 278	1 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   ∖ cdif 3902   ⊆ wss 3905  {csn 4589  ran crn 5662  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  GIdcgi 30851  RingOpscrngo 38573  DivRingOpscdrng 38627

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-1st 7982  df-2nd 7983  df-1o 8449  df-en 8940  df-grpo 30854  df-gid 30855  df-ginv 30856  df-ablo 30906  df-ass 38522  df-exid 38524  df-mgmOLD 38528  df-sgrOLD 38540  df-mndo 38546  df-rngo 38574  df-drngo 38628

This theorem is used by:  isfldidl  38747