Theorem divrngidl 34784

Description: The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

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

Assertion

Ref	Expression
divrngidl	⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

Proof of Theorem divrngidl

Dummy variables 𝑖 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		divrngidl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		divrngidl.2	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
3		divrngidl.4	. . 3 ⊢ 𝑍 = (GId‘𝐺)
4		divrngidl.3	. . 3 ⊢ 𝑋 = ran 𝐺
5		eqid 2793	. . 3 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdrngo2 34714	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
7	1, 3	idl0cl 34774	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑖)
8	7	adantr 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑍 ∈ 𝑖)
9	3	fvexi 6544	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
10	9	snss 4619	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ 𝑖 ↔ {𝑍} ⊆ 𝑖)
11		necom 3035	. . . . . . . . . . . 12 ⊢ (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
12		pssdifn0 4239	. . . . . . . . . . . . 13 ⊢ (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
13		n0 4224	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
14	12, 13	sylib 219	. . . . . . . . . . . 12 ⊢ (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
15	10, 11, 14	syl2anb 597	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ 𝑖 ∧ 𝑖 ≠ {𝑍}) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
16	1, 4	idlss 34772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ 𝑋)
17		ssdif 4032	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ⊆ 𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
18	17	sselda 3884	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
19	16, 18	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
20		oveq2 7015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
21	20	eqeq1d 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
22	21	rexbidv 3257	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
23	22	rspcva 3552	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
24	19, 23	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
25		eldifi 4019	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧 ∈ 𝑖)
26		eldifi 4019	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑍}) → 𝑦 ∈ 𝑋)
27	25, 26	anim12i 612	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋))
28	1, 2, 4	idllmulcl 34776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
29	1, 2, 4, 5	1idl 34782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖 ↔ 𝑖 = 𝑋))
30	29	biimpd 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋))
31	30	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋))
32		eleq1 2868	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
33	32	imbi1d 343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖 → 𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋)))
34	31, 33	syl5ibrcom 248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 → 𝑖 = 𝑋)))
35	28, 34	mpid 44	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
36	27, 35	sylan2 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
37	36	anassrs 468	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
38	37	rexlimdva 3244	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
39	38	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)) → 𝑖 = 𝑋)
40	24, 39	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑖 = 𝑋)
41	40	an32s 648	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑖 = 𝑋)
42	41	ex 413	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
43	42	exlimdv 1909	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
44	15, 43	syl5 34	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑍 ∈ 𝑖 ∧ 𝑖 ≠ {𝑍}) → 𝑖 = 𝑋))
45	8, 44	mpand 691	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
46	45	an32s 648	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
47		neor 3074	. . . . . . . 8 ⊢ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ↔ (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
48	46, 47	sylibr 235	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
49	48	ex 413	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
50	1, 3	0idl 34781	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
51		eleq1 2868	. . . . . . . . 9 ⊢ (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
52	50, 51	syl5ibrcom 248	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
53	1, 4	rngoidl 34780	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
54		eleq1 2868	. . . . . . . . 9 ⊢ (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
55	53, 54	syl5ibrcom 248	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = 𝑋 → 𝑖 ∈ (Idl‘𝑅)))
56	52, 55	jaod 854	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
57	56	adantr 481	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
58	49, 57	impbid 213	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
59		vex 3435	. . . . . 6 ⊢ 𝑖 ∈ V
60	59	elpr 4489	. . . . 5 ⊢ (𝑖 ∈ {{𝑍}, 𝑋} ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
61	58, 60	syl6bbr 290	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ {{𝑍}, 𝑋}))
62	61	eqrdv 2791	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (Idl‘𝑅) = {{𝑍}, 𝑋})
63	62	adantrl 712	. 2 ⊢ ((𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))) → (Idl‘𝑅) = {{𝑍}, 𝑋})
64	6, 63	sylbi 218	1 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 842   = wceq 1520  ∃wex 1759   ∈ wcel 2079   ≠ wne 2982  ∀wral 3103  ∃wrex 3104   ∖ cdif 3851   ⊆ wss 3854  ∅c0 4206  {csn 4466  {cpr 4468  ran crn 5436  ‘cfv 6217  (class class class)co 7007  1^st c1st 7534  2^nd c2nd 7535  GIdcgi 27946  RingOpscrngo 34650  DivRingOpscdrng 34704  Idlcidl 34763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-om 7428  df-1st 7536  df-2nd 7537  df-1o 7944  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-grpo 27949  df-gid 27950  df-ginv 27951  df-ablo 28001  df-ass 34599  df-exid 34601  df-mgmOLD 34605  df-sgrOLD 34617  df-mndo 34623  df-rngo 34651  df-drngo 34705  df-idl 34766

This theorem is referenced by:  divrngpr  34809  isfldidl  34824