Theorem divrngidl 38047

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 2730	. . 3 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdrngo2 37977	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
7	1, 3	idl0cl 38037	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑖)
8	7	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑍 ∈ 𝑖)
9	3	fvexi 6831	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
10	9	snss 4735	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ 𝑖 ↔ {𝑍} ⊆ 𝑖)
11		necom 2979	. . . . . . . . . . . 12 ⊢ (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
12		pssdifn0 4316	. . . . . . . . . . . . 13 ⊢ (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
13		n0 4301	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
14	12, 13	sylib 218	. . . . . . . . . . . 12 ⊢ (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
15	10, 11, 14	syl2anb 598	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ 𝑖 ∧ 𝑖 ≠ {𝑍}) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
16	1, 4	idlss 38035	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ 𝑋)
17		ssdif 4092	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ⊆ 𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
18	17	sselda 3932	. . . . . . . . . . . . . . . . 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 7349	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
21	20	eqeq1d 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
22	21	rexbidv 3154	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
23	22	rspcva 3573	. . . . . . . . . . . . . . . 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 4079	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧 ∈ 𝑖)
26		eldifi 4079	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑍}) → 𝑦 ∈ 𝑋)
27	25, 26	anim12i 613	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋))
28	1, 2, 4	idllmulcl 38039	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
29	1, 2, 4, 5	1idl 38045	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖 ↔ 𝑖 = 𝑋))
30	29	biimpd 229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋))
31	30	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋))
32		eleq1 2817	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
33	32	imbi1d 341	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖 → 𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋)))
34	31, 33	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
37	36	anassrs 467	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
38	37	rexlimdva 3131	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
39	38	imp 406	. . . . . . . . . . . . . . 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 652	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑖 = 𝑋)
42	41	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
43	42	exlimdv 1934	. . . . . . . . . . 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 695	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
46	45	an32s 652	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
47		neor 3018	. . . . . . . 8 ⊢ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ↔ (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
48	46, 47	sylibr 234	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
49	48	ex 412	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
50	1, 3	0idl 38044	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
51		eleq1 2817	. . . . . . . . 9 ⊢ (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
52	50, 51	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
53	1, 4	rngoidl 38043	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
54		eleq1 2817	. . . . . . . . 9 ⊢ (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
55	53, 54	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = 𝑋 → 𝑖 ∈ (Idl‘𝑅)))
56	52, 55	jaod 859	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
57	56	adantr 480	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
58	49, 57	impbid 212	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
59		vex 3438	. . . . . 6 ⊢ 𝑖 ∈ V
60	59	elpr 4599	. . . . 5 ⊢ (𝑖 ∈ {{𝑍}, 𝑋} ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
61	58, 60	bitr4di 289	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ {{𝑍}, 𝑋}))
62	61	eqrdv 2728	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (Idl‘𝑅) = {{𝑍}, 𝑋})
63	62	adantrl 716	. 2 ⊢ ((𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))) → (Idl‘𝑅) = {{𝑍}, 𝑋})
64	6, 63	sylbi 217	1 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2110   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ∖ cdif 3897   ⊆ wss 3900  ∅c0 4281  {csn 4574  {cpr 4576  ran crn 5615  ‘cfv 6477  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  GIdcgi 30460  RingOpscrngo 37913  DivRingOpscdrng 37967  Idlcidl 38026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-1st 7916  df-2nd 7917  df-1o 8380  df-en 8865  df-grpo 30463  df-gid 30464  df-ginv 30465  df-ablo 30515  df-ass 37862  df-exid 37864  df-mgmOLD 37868  df-sgrOLD 37880  df-mndo 37886  df-rngo 37914  df-drngo 37968  df-idl 38029

This theorem is referenced by:  divrngpr  38072  isfldidl  38087