prnc - Mathbox for Jeff Madsen

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Theorem prnc 33036

Description: A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

**Hypotheses**
Ref	Expression
prnc.1	⊢ 𝐺 = (1^st ‘𝑅)
prnc.2	⊢ 𝐻 = (2^nd ‘𝑅)
prnc.3	⊢ 𝑋 = ran 𝐺

**Assertion**
Ref	Expression
prnc	⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})

Distinct variable groups: 𝑥,𝑅,𝑦 𝑥,𝑋,𝑦 𝑥,𝐺,𝑦 𝑥,𝐻,𝑦 𝑥,𝐴,𝑦

Proof of Theorem prnc Dummy variables 𝑗 𝑢 𝑣 𝑤 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngorngo 32969	. . . . 5 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2		ssrab2 3650	. . . . . . 7 ⊢ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋
3	2	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋)
4		prnc.1	. . . . . . . . 9 ⊢ 𝐺 = (1^st ‘𝑅)
5		prnc.3	. . . . . . . . 9 ⊢ 𝑋 = ran 𝐺
6		eqid 2610	. . . . . . . . 9 ⊢ (GId‘𝐺) = (GId‘𝐺)
7	4, 5, 6	rngo0cl 32888	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
8	7	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (GId‘𝐺) ∈ 𝑋)
9		prnc.2	. . . . . . . . . 10 ⊢ 𝐻 = (2^nd ‘𝑅)
10	6, 5, 4, 9	rngolz 32891	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
11	10	eqcomd 2616	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (GId‘𝐺) = ((GId‘𝐺)𝐻𝐴))
12		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑦 = (GId‘𝐺) → (𝑦𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
13	12	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑦 = (GId‘𝐺) → ((GId‘𝐺) = (𝑦𝐻𝐴) ↔ (GId‘𝐺) = ((GId‘𝐺)𝐻𝐴)))
14	13	rspcev 3282	. . . . . . . 8 ⊢ (((GId‘𝐺) ∈ 𝑋 ∧ (GId‘𝐺) = ((GId‘𝐺)𝐻𝐴)) → ∃𝑦 ∈ 𝑋 (GId‘𝐺) = (𝑦𝐻𝐴))
15	8, 11, 14	syl2anc 691	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (GId‘𝐺) = (𝑦𝐻𝐴))
16		eqeq1 2614	. . . . . . . . 9 ⊢ (𝑥 = (GId‘𝐺) → (𝑥 = (𝑦𝐻𝐴) ↔ (GId‘𝐺) = (𝑦𝐻𝐴)))
17	16	rexbidv 3034	. . . . . . . 8 ⊢ (𝑥 = (GId‘𝐺) → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦 ∈ 𝑋 (GId‘𝐺) = (𝑦𝐻𝐴)))
18	17	elrab 3331	. . . . . . 7 ⊢ ((GId‘𝐺) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ((GId‘𝐺) ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 (GId‘𝐺) = (𝑦𝐻𝐴)))
19	8, 15, 18	sylanbrc 695	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (GId‘𝐺) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
20		eqeq1 2614	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝑥 = (𝑦𝐻𝐴) ↔ 𝑢 = (𝑦𝐻𝐴)))
21	20	rexbidv 3034	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦 ∈ 𝑋 𝑢 = (𝑦𝐻𝐴)))
22		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑟 → (𝑦𝐻𝐴) = (𝑟𝐻𝐴))
23	22	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝑢 = (𝑦𝐻𝐴) ↔ 𝑢 = (𝑟𝐻𝐴)))
24	23	cbvrexv 3148	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝑋 𝑢 = (𝑦𝐻𝐴) ↔ ∃𝑟 ∈ 𝑋 𝑢 = (𝑟𝐻𝐴))
25	21, 24	syl6bb 275	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑟 ∈ 𝑋 𝑢 = (𝑟𝐻𝐴)))
26	25	elrab 3331	. . . . . . . 8 ⊢ (𝑢 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝑢 ∈ 𝑋 ∧ ∃𝑟 ∈ 𝑋 𝑢 = (𝑟𝐻𝐴)))
27		eqeq1 2614	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → (𝑥 = (𝑦𝐻𝐴) ↔ 𝑣 = (𝑦𝐻𝐴)))
28	27	rexbidv 3034	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦 ∈ 𝑋 𝑣 = (𝑦𝐻𝐴)))
29		oveq1 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑠 → (𝑦𝐻𝐴) = (𝑠𝐻𝐴))
30	29	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑠 → (𝑣 = (𝑦𝐻𝐴) ↔ 𝑣 = (𝑠𝐻𝐴)))
31	30	cbvrexv 3148	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦 ∈ 𝑋 𝑣 = (𝑦𝐻𝐴) ↔ ∃𝑠 ∈ 𝑋 𝑣 = (𝑠𝐻𝐴))
32	28, 31	syl6bb 275	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑠 ∈ 𝑋 𝑣 = (𝑠𝐻𝐴)))
33	32	elrab 3331	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝑣 ∈ 𝑋 ∧ ∃𝑠 ∈ 𝑋 𝑣 = (𝑠𝐻𝐴)))
34	4, 9, 5	rngodir 32874	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ RingOps ∧ (𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)))
35	34	3exp2 1277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ RingOps → (𝑟 ∈ 𝑋 → (𝑠 ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴))))))
36	35	imp42 618	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ (𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)))
37	4, 5	rngogcl 32881	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑟𝐺𝑠) ∈ 𝑋)
38	37	3expib 1260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ RingOps → ((𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑟𝐺𝑠) ∈ 𝑋))
39	38	imdistani 722	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ RingOps ∧ (𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) → (𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋))
40	4, 9, 5	rngocl 32870	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ 𝑋)
41	40	3expa 1257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ 𝑋)
42		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴)
43		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = (𝑟𝐺𝑠) → (𝑦𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴))
44	43	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = (𝑟𝐺𝑠) → (((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴) ↔ ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴)))
45	44	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑟𝐺𝑠) ∈ 𝑋 ∧ ((𝑟𝐺𝑠)𝐻𝐴) = ((𝑟𝐺𝑠)𝐻𝐴)) → ∃𝑦 ∈ 𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴))
46	42, 45	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑟𝐺𝑠) ∈ 𝑋 → ∃𝑦 ∈ 𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴))
47	46	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴))
48		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ((𝑟𝐺𝑠)𝐻𝐴) → (𝑥 = (𝑦𝐻𝐴) ↔ ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴)))
49	48	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ((𝑟𝐺𝑠)𝐻𝐴) → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦 ∈ 𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴)))
50	49	elrab 3331	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑟𝐺𝑠)𝐻𝐴) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (((𝑟𝐺𝑠)𝐻𝐴) ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 ((𝑟𝐺𝑠)𝐻𝐴) = (𝑦𝐻𝐴)))
51	41, 47, 50	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ RingOps ∧ (𝑟𝐺𝑠) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
52	39, 51	sylan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ RingOps ∧ (𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑟𝐺𝑠)𝐻𝐴) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
53	36, 52	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ RingOps ∧ (𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
54	53	an32s 842	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑟 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) → ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
55	54	anassrs 678	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ 𝑠 ∈ 𝑋) → ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
56		oveq2 6557	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝑠𝐻𝐴) → ((𝑟𝐻𝐴)𝐺𝑣) = ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)))
57	56	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑠𝐻𝐴) → (((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ((𝑟𝐻𝐴)𝐺(𝑠𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
58	55, 57	syl5ibrcom 236	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ 𝑠 ∈ 𝑋) → (𝑣 = (𝑠𝐻𝐴) → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
59	58	rexlimdva 3013	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) → (∃𝑠 ∈ 𝑋 𝑣 = (𝑠𝐻𝐴) → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
60	59	adantld 482	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) → ((𝑣 ∈ 𝑋 ∧ ∃𝑠 ∈ 𝑋 𝑣 = (𝑠𝐻𝐴)) → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
61	33, 60	syl5bi 231	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) → (𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} → ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
62	61	ralrimiv 2948	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) → ∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
63	4, 9, 5	rngoass 32875	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ RingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴)))
64	63	3exp2 1277	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ RingOps → (𝑤 ∈ 𝑋 → (𝑟 ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴))))))
65	64	imp42 618	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴)))
66	65	an32s 842	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → ((𝑤𝐻𝑟)𝐻𝐴) = (𝑤𝐻(𝑟𝐻𝐴)))
67	4, 9, 5	rngocl 32870	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ RingOps ∧ 𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋) → (𝑤𝐻𝑟) ∈ 𝑋)
68	67	3expib 1260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ RingOps → ((𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋) → (𝑤𝐻𝑟) ∈ 𝑋))
69	68	imdistani 722	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ RingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → (𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋))
70	4, 9, 5	rngocl 32870	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ 𝑋)
71	70	3expa 1257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ 𝑋)
72		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤𝐻𝑟)𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴)
73		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑤𝐻𝑟) → (𝑦𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴))
74	73	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑤𝐻𝑟) → (((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴) ↔ ((𝑤𝐻𝑟)𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴)))
75	74	rspcev 3282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤𝐻𝑟) ∈ 𝑋 ∧ ((𝑤𝐻𝑟)𝐻𝐴) = ((𝑤𝐻𝑟)𝐻𝐴)) → ∃𝑦 ∈ 𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴))
76	72, 75	mpan2 703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤𝐻𝑟) ∈ 𝑋 → ∃𝑦 ∈ 𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴))
77	76	ad2antlr 759	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴))
78		eqeq1 2614	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ((𝑤𝐻𝑟)𝐻𝐴) → (𝑥 = (𝑦𝐻𝐴) ↔ ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴)))
79	78	rexbidv 3034	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑤𝐻𝑟)𝐻𝐴) → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦 ∈ 𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴)))
80	79	elrab 3331	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (((𝑤𝐻𝑟)𝐻𝐴) ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 ((𝑤𝐻𝑟)𝐻𝐴) = (𝑦𝐻𝐴)))
81	71, 77, 80	sylanbrc 695	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ RingOps ∧ (𝑤𝐻𝑟) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
82	69, 81	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
83	82	an32s 842	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → ((𝑤𝐻𝑟)𝐻𝐴) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
84	66, 83	eqeltrrd 2689	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑤 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
85	84	anass1rs 845	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
86	85	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) → ∀𝑤 ∈ 𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
87	62, 86	jca 553	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) → (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
88		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑟𝐻𝐴) → (𝑢𝐺𝑣) = ((𝑟𝐻𝐴)𝐺𝑣))
89	88	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑟𝐻𝐴) → ((𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
90	89	ralbidv 2969	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑟𝐻𝐴) → (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
91		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑟𝐻𝐴) → (𝑤𝐻𝑢) = (𝑤𝐻(𝑟𝐻𝐴)))
92	91	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑟𝐻𝐴) → ((𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
93	92	ralbidv 2969	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑟𝐻𝐴) → (∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ∀𝑤 ∈ 𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
94	90, 93	anbi12d 743	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑟𝐻𝐴) → ((∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}) ↔ (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ((𝑟𝐻𝐴)𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻(𝑟𝐻𝐴)) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})))
95	87, 94	syl5ibrcom 236	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) → (𝑢 = (𝑟𝐻𝐴) → (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})))
96	95	rexlimdva 3013	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (∃𝑟 ∈ 𝑋 𝑢 = (𝑟𝐻𝐴) → (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})))
97	96	adantld 482	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑢 ∈ 𝑋 ∧ ∃𝑟 ∈ 𝑋 𝑢 = (𝑟𝐻𝐴)) → (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})))
98	26, 97	syl5bi 231	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} → (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})))
99	98	ralrimiv 2948	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∀𝑢 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))
100	3, 19, 99	3jca 1235	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})))
101	1, 100	sylan 487	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})))
102	4, 9, 5, 6	isidlc 32984	. . . . 5 ⊢ (𝑅 ∈ CRingOps → ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ↔ ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))))
103	102	adantr 480	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ↔ ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑋 ∧ (GId‘𝐺) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑢 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (∀𝑣 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} (𝑢𝐺𝑣) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑤 ∈ 𝑋 (𝑤𝐻𝑢) ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}))))
104	101, 103	mpbird 246	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅))
105		simpr 476	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
106	4	rneqi 5273	. . . . . . . . . 10 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
107	5, 106	eqtri 2632	. . . . . . . . 9 ⊢ 𝑋 = ran (1^st ‘𝑅)
108		eqid 2610	. . . . . . . . 9 ⊢ (GId‘𝐻) = (GId‘𝐻)
109	107, 9, 108	rngo1cl 32908	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
110	109	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (GId‘𝐻) ∈ 𝑋)
111	9, 107, 108	rngolidm 32906	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐻)𝐻𝐴) = 𝐴)
112	111	eqcomd 2616	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 = ((GId‘𝐻)𝐻𝐴))
113		oveq1 6556	. . . . . . . . 9 ⊢ (𝑦 = (GId‘𝐻) → (𝑦𝐻𝐴) = ((GId‘𝐻)𝐻𝐴))
114	113	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑦 = (GId‘𝐻) → (𝐴 = (𝑦𝐻𝐴) ↔ 𝐴 = ((GId‘𝐻)𝐻𝐴)))
115	114	rspcev 3282	. . . . . . 7 ⊢ (((GId‘𝐻) ∈ 𝑋 ∧ 𝐴 = ((GId‘𝐻)𝐻𝐴)) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦𝐻𝐴))
116	110, 112, 115	syl2anc 691	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦𝐻𝐴))
117	1, 116	sylan 487	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦𝐻𝐴))
118		eqeq1 2614	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = (𝑦𝐻𝐴) ↔ 𝐴 = (𝑦𝐻𝐴)))
119	118	rexbidv 3034	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦𝐻𝐴)))
120	119	elrab 3331	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ (𝐴 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦𝐻𝐴)))
121	105, 117, 120	sylanbrc 695	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
122	121	snssd 4281	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
123		snssg 4268	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑗 ↔ {𝐴} ⊆ 𝑗))
124	123	biimpar 501	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝐴} ⊆ 𝑗) → 𝐴 ∈ 𝑗)
125	4, 9, 5	idllmulcl 32989	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝑗 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝐻𝐴) ∈ 𝑗)
126	125	anassrs 678	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝑗) ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝐴) ∈ 𝑗)
127		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦𝐻𝐴) → (𝑥 ∈ 𝑗 ↔ (𝑦𝐻𝐴) ∈ 𝑗))
128	126, 127	syl5ibrcom 236	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝑗) ∧ 𝑦 ∈ 𝑋) → (𝑥 = (𝑦𝐻𝐴) → 𝑥 ∈ 𝑗))
129	128	rexlimdva 3013	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝑗) → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥 ∈ 𝑗))
130	129	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝑗) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥 ∈ 𝑗))
131	130	ralrimiva 2949	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝑗) → ∀𝑥 ∈ 𝑋 (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥 ∈ 𝑗))
132		rabss 3642	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗 ↔ ∀𝑥 ∈ 𝑋 (∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴) → 𝑥 ∈ 𝑗))
133	131, 132	sylibr 223	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝑗) → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗)
134	133	ex 449	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝐴 ∈ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
135	124, 134	syl5 33	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → ((𝐴 ∈ 𝑋 ∧ {𝐴} ⊆ 𝑗) → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
136	135	expdimp 452	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝑋) → ({𝐴} ⊆ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
137	136	an32s 842	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑗 ∈ (Idl‘𝑅)) → ({𝐴} ⊆ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
138	137	ralrimiva 2949	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
139	1, 138	sylan 487	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))
140	104, 122, 139	3jca 1235	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ∧ {𝐴} ⊆ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗)))
141		snssi 4280	. . 3 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ⊆ 𝑋)
142	4, 5	igenval2 33035	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ {𝐴} ⊆ 𝑋) → ((𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ∧ {𝐴} ⊆ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))))
143	1, 141, 142	syl2an 493	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → ((𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ↔ ({𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∈ (Idl‘𝑅) ∧ {𝐴} ⊆ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ∧ ∀𝑗 ∈ (Idl‘𝑅)({𝐴} ⊆ 𝑗 → {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)} ⊆ 𝑗))))
144	140, 143	mpbird 246	1 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 ⊆ wss 3540 {csn 4125 ran crn 5039 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 GIdcgi 26728 RingOpscrngo 32863 CRingOpsccring 32962 Idlcidl 32976 IdlGen cigen 33028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-grpo 26731 df-gid 26732 df-ginv 26733 df-ablo 26783 df-ass 32812 df-exid 32814 df-mgmOLD 32818 df-sgrOLD 32830 df-mndo 32836 df-rngo 32864 df-com2 32959 df-crngo 32963 df-idl 32979 df-igen 33029

This theorem is referenced by: isfldidl 33037 ispridlc 33039

W3C validator