Theorem divrngidl 37200

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 2731	. . 3 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdrngo2 37130	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
7	1, 3	idl0cl 37190	. . . . . . . . . . 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 6906	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
10	9	snss 4790	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ 𝑖 ↔ {𝑍} ⊆ 𝑖)
11		necom 2993	. . . . . . . . . . . 12 ⊢ (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
12		pssdifn0 4366	. . . . . . . . . . . . 13 ⊢ (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
13		n0 4347	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
14	12, 13	sylib 217	. . . . . . . . . . . 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 37188	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ 𝑋)
17		ssdif 4140	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ⊆ 𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
18	17	sselda 3983	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
19	16, 18	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
20		oveq2 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
21	20	eqeq1d 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
22	21	rexbidv 3177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
23	22	rspcva 3611	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
24	19, 23	sylan 579	. . . . . . . . . . . . . . 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 4127	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧 ∈ 𝑖)
26		eldifi 4127	. . . . . . . . . . . . . . . . . . . 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 37192	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
29	1, 2, 4, 5	1idl 37198	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖 ↔ 𝑖 = 𝑋))
30	29	biimpd 228	. . . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
33	32	imbi1d 340	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖 → 𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋)))
34	31, 33	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
38	37	rexlimdva 3154	. . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑖 = 𝑋)
41	40	an32s 649	. . . . . . . . . . . . 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 1935	. . . . . . . . . . 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 692	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
46	45	an32s 649	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
47		neor 3033	. . . . . . . 8 ⊢ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ↔ (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
48	46, 47	sylibr 233	. . . . . . 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 37197	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
51		eleq1 2820	. . . . . . . . 9 ⊢ (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
52	50, 51	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
53	1, 4	rngoidl 37196	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
54		eleq1 2820	. . . . . . . . 9 ⊢ (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
55	53, 54	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = 𝑋 → 𝑖 ∈ (Idl‘𝑅)))
56	52, 55	jaod 856	. . . . . . 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 211	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
59		vex 3477	. . . . . 6 ⊢ 𝑖 ∈ V
60	59	elpr 4652	. . . . 5 ⊢ (𝑖 ∈ {{𝑍}, 𝑋} ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
61	58, 60	bitr4di 288	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ {{𝑍}, 𝑋}))
62	61	eqrdv 2729	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (Idl‘𝑅) = {{𝑍}, 𝑋})
63	62	adantrl 713	. 2 ⊢ ((𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))) → (Idl‘𝑅) = {{𝑍}, 𝑋})
64	6, 63	sylbi 216	1 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323  {csn 4629  {cpr 4631  ran crn 5678  ‘cfv 6544  (class class class)co 7412  1^st c1st 7976  2^nd c2nd 7977  GIdcgi 30007  RingOpscrngo 37066  DivRingOpscdrng 37120  Idlcidl 37179

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-1st 7978  df-2nd 7979  df-1o 8469  df-en 8943  df-grpo 30010  df-gid 30011  df-ginv 30012  df-ablo 30062  df-ass 37015  df-exid 37017  df-mgmOLD 37021  df-sgrOLD 37033  df-mndo 37039  df-rngo 37067  df-drngo 37121  df-idl 37182

This theorem is referenced by:  divrngpr  37225  isfldidl  37240