Theorem divrngidl 39005

Description: Obsolete theorem, use drngnidl 21531 instead. The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 2760	. . 3 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdrngo2 38935	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
7	1, 3	idl0cl 38995	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑖)
8	7	adantr 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑍 ∈ 𝑖)
9	3	fvexi 6899	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
10	9	snss 4744	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ 𝑖 ↔ {𝑍} ⊆ 𝑖)
11		necom 3008	. . . . . . . . . . . 12 ⊢ (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
12		pssdifn0 4315	. . . . . . . . . . . . 13 ⊢ (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
13		n0 4299	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
14	12, 13	sylib 221	. . . . . . . . . . . 12 ⊢ (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
15	10, 11, 14	syl2anb 610	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ 𝑖 ∧ 𝑖 ≠ {𝑍}) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
16	1, 4	idlss 38993	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ 𝑋)
17		ssdif 4080	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ⊆ 𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
18	17	sselda 3908	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
19	16, 18	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
20		oveq2 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
21	20	eqeq1d 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
22	21	rexbidv 3186	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
23	22	rspcva 3574	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
24	19, 23	sylan 592	. . . . . . . . . . . . . . 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 4067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧 ∈ 𝑖)
26		eldifi 4067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑍}) → 𝑦 ∈ 𝑋)
27	25, 26	anim12i 625	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋))
28	1, 2, 4	idllmulcl 38997	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
29	1, 2, 4, 5	1idl 39003	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖 ↔ 𝑖 = 𝑋))
30	29	biimpd 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋))
31	30	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋))
32		eleq1 2848	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
33	32	imbi1d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖 → 𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖 → 𝑖 = 𝑋)))
34	31, 33	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 → 𝑖 = 𝑋)))
35	28, 34	mpid 45	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ 𝑖 ∧ 𝑦 ∈ 𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
36	27, 35	sylan2 605	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
37	36	anassrs 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
38	37	rexlimdva 3163	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
39	38	imp 412	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)) → 𝑖 = 𝑋)
40	24, 39	syldan 603	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑖 = 𝑋)
41	40	an32s 665	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑖 = 𝑋)
42	41	ex 418	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
43	42	exlimdv 1965	. . . . . . . . . . 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 35	. . . . . . . . . 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 708	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
46	45	an32s 665	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
47		neor 3047	. . . . . . . 8 ⊢ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ↔ (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
48	46, 47	sylibr 237	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
49	48	ex 418	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
50	1, 3	0idl 39002	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
51		eleq1 2848	. . . . . . . . 9 ⊢ (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
52	50, 51	syl5ibrcom 250	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
53	1, 4	rngoidl 39001	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
54		eleq1 2848	. . . . . . . . 9 ⊢ (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
55	53, 54	syl5ibrcom 250	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑖 = 𝑋 → 𝑖 ∈ (Idl‘𝑅)))
56	52, 55	jaod 873	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
57	56	adantr 486	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
58	49, 57	impbid 215	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
59		vex 3454	. . . . . 6 ⊢ 𝑖 ∈ V
60	59	elpr 4608	. . . . 5 ⊢ (𝑖 ∈ {{𝑍}, 𝑋} ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
61	58, 60	bitr4di 292	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ {{𝑍}, 𝑋}))
62	61	eqrdv 2758	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (Idl‘𝑅) = {{𝑍}, 𝑋})
63	62	adantrl 729	. 2 ⊢ ((𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))) → (Idl‘𝑅) = {{𝑍}, 𝑋})
64	6, 63	sylbi 220	1 ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  {csn 4583  {cpr 4585  ran crn 5651  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  GIdcgi 31092  RingOpscrngo 38871  DivRingOpscdrng 38925  Idlcidl 38984

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-1st 8001  df-2nd 8002  df-1o 8476  df-en 8974  df-grpo 31095  df-gid 31096  df-ginv 31097  df-ablo 31147  df-ass 38820  df-exid 38822  df-mgmOLD 38826  df-sgrOLD 38838  df-mndo 38844  df-rngo 38872  df-drngo 38926  df-idl 38987

This theorem is used by:  divrngpr  39030  isfldidl  39045