Theorem rlocf1 33455

Description: The embedding 𝐹 of a ring 𝑅 into its localization 𝐿. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
rlocf1.1	⊢ 𝐵 = (Base‘𝑅)
rlocf1.2	⊢ 1 = (1r‘𝑅)
rlocf1.3	⊢ 𝐿 = (𝑅 RLocal 𝑆)
rlocf1.4	⊢ ∼ = (𝑅 ~RL 𝑆)
rlocf1.5	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )
rlocf1.6	⊢ (𝜑 → 𝑅 ∈ CRing)
rlocf1.7	⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
rlocf1.8	⊢ (𝜑 → 𝑆 ⊆ (RLReg‘𝑅))

Assertion

Ref	Expression
rlocf1	⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))

Distinct variable groups:   𝑥, 1   𝑥, ∼   𝑥,𝐵   𝑥,𝐹   𝑥,𝐿   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥

Proof of Theorem rlocf1

Dummy variables 𝑡 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
2		rlocf1.7	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
3		eqid 2762	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4		rlocf1.2	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
5	3, 4	ringidval 20233	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘𝑅))
6	5	subm0cl 18845	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
7	2, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝑆)
8	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ 𝑆)
9	1, 8	opelxpd 5686	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
10		rlocf1.4	. . . . . . 7 ⊢ ∼ = (𝑅 ~RL 𝑆)
11	10	ovexi 7430	. . . . . 6 ⊢ ∼ ∈ V
12	11	ecelqsi 8751	. . . . 5 ⊢ (⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆) → [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
13	9, 12	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
14	13	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
15		rlocf1.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing)
16	15	crnggrpd 20297	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
17	16	ad5antr 744	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑅 ∈ Grp)
18		simp-5r 795	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑥 ∈ 𝐵)
19		simp-4r 793	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑦 ∈ 𝐵)
20		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
21	4	fvexi 6881	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
22	20, 21	op1st 7978	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨𝑥, 1 ⟩) = 𝑥
23	22	a1i 11	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (1st ‘⟨𝑥, 1 ⟩) = 𝑥)
24		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
25	24, 21	op2nd 7979	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑦, 1 ⟩) = 1
26	25	a1i 11	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (2nd ‘⟨𝑦, 1 ⟩) = 1 )
27	23, 26	oveq12d 7414	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → ((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩)) = (𝑥(.r‘𝑅) 1 ))
28		rlocf1.1	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑅)
29		eqid 2762	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
30	15	crngringd 20296	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
31	30	ad5antr 744	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑅 ∈ Ring)
32	28, 29, 4, 31, 18	ringridmd 20323	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (𝑥(.r‘𝑅) 1 ) = 𝑥)
33	27, 32	eqtrd 2797	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → ((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩)) = 𝑥)
34	24, 21	op1st 7978	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨𝑦, 1 ⟩) = 𝑦
35	34	a1i 11	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (1st ‘⟨𝑦, 1 ⟩) = 𝑦)
36	20, 21	op2nd 7979	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑥, 1 ⟩) = 1
37	36	a1i 11	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (2nd ‘⟨𝑥, 1 ⟩) = 1 )
38	35, 37	oveq12d 7414	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → ((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)) = (𝑦(.r‘𝑅) 1 ))
39	28, 29, 4, 31, 19	ringridmd 20323	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (𝑦(.r‘𝑅) 1 ) = 𝑦)
40	38, 39	eqtrd 2797	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → ((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)) = 𝑦)
41	33, 40	oveq12d 7414	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) = (𝑥(-g‘𝑅)𝑦))
42		rlocf1.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (RLReg‘𝑅))
43	42	ad5antr 744	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑆 ⊆ (RLReg‘𝑅))
44		simplr 778	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑡 ∈ 𝑆)
45	43, 44	sseldd 3937	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑡 ∈ (RLReg‘𝑅))
46	23, 18	eqeltrd 2862	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (1st ‘⟨𝑥, 1 ⟩) ∈ 𝐵)
47	3, 28	mgpbas 20191	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
48	47	submss 18843	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
49	2, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
50	49, 7	sseldd 3937	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ 𝐵)
51	50	ad5antr 744	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 1 ∈ 𝐵)
52	26, 51	eqeltrd 2862	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (2nd ‘⟨𝑦, 1 ⟩) ∈ 𝐵)
53	28, 29, 31, 46, 52	ringcld 20310	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → ((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩)) ∈ 𝐵)
54	35, 19	eqeltrd 2862	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (1st ‘⟨𝑦, 1 ⟩) ∈ 𝐵)
55	37, 51	eqeltrd 2862	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (2nd ‘⟨𝑥, 1 ⟩) ∈ 𝐵)
56	28, 29, 31, 54, 55	ringcld 20310	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → ((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)) ∈ 𝐵)
57		eqid 2762	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
58	28, 57	grpsubcl 19062	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ ((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩)) ∈ 𝐵 ∧ ((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)) ∈ 𝐵) → (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) ∈ 𝐵)
59	17, 53, 56, 58	syl3anc 1390	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) ∈ 𝐵)
60		simpr 488	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅))
61		eqid 2762	. . . . . . . . . . . 12 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
62		eqid 2762	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
63	61, 28, 29, 62	rrgeq0i 20749	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (RLReg‘𝑅) ∧ (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) ∈ 𝐵) → ((𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅) → (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) = (0g‘𝑅)))
64	63	imp 410	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (RLReg‘𝑅) ∧ (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) ∈ 𝐵) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) = (0g‘𝑅))
65	45, 59, 60, 64	syl21anc 848	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩))) = (0g‘𝑅))
66	41, 65	eqtr3d 2799	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → (𝑥(-g‘𝑅)𝑦) = (0g‘𝑅))
67	28, 62, 57	grpsubeq0 19068	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-g‘𝑅)𝑦) = (0g‘𝑅) ↔ 𝑥 = 𝑦))
68	67	biimpa 480	. . . . . . . 8 ⊢ (((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-g‘𝑅)𝑦) = (0g‘𝑅)) → 𝑥 = 𝑦)
69	17, 18, 19, 66, 68	syl31anc 1392	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅)) → 𝑥 = 𝑦)
70	49	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑆 ⊆ 𝐵)
71		eqid 2762	. . . . . . . . . . 11 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
72	15	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
73	2	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
74	28, 62, 4, 29, 57, 71, 10, 72, 73	erler 33446	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ∼ Er (𝐵 × 𝑆))
75	9	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
76	74, 75	erth 8733	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩ ↔ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ))
77	76	biimpar 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → ⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩)
78	28, 10, 70, 62, 29, 57, 77	erldi 33443	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘⟨𝑥, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑦, 1 ⟩))(-g‘𝑅)((1st ‘⟨𝑦, 1 ⟩)(.r‘𝑅)(2nd ‘⟨𝑥, 1 ⟩)))) = (0g‘𝑅))
79	69, 78	r19.29a 3170	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑥 = 𝑦)
80	79	ex 416	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
81	80	anasss 470	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
82	81	ralrimivva 3205	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
83		rlocf1.5	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )
84		opeq1 4831	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 1 ⟩ = ⟨𝑦, 1 ⟩)
85	84	eceq1d 8719	. . . 4 ⊢ (𝑥 = 𝑦 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ )
86	83, 85	f1mpt 7245	. . 3 ⊢ (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ↔ (∀𝑥 ∈ 𝐵 [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦)))
87	14, 82, 86	sylanbrc 592	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ))
88		eqid 2762	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
89		eqid 2762	. . 3 ⊢ (.r‘𝐿) = (.r‘𝐿)
90		eqid 2762	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
91		rlocf1.3	. . . . 5 ⊢ 𝐿 = (𝑅 RLocal 𝑆)
92	28, 29, 90, 91, 10, 15, 2	rloccring 33452	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
93	92	crngringd 20296	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
94		opeq1 4831	. . . . . 6 ⊢ (𝑥 = 1 → ⟨𝑥, 1 ⟩ = ⟨ 1 , 1 ⟩)
95	94	eceq1d 8719	. . . . 5 ⊢ (𝑥 = 1 → [⟨𝑥, 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼ )
96		eqid 2762	. . . . . 6 ⊢ [⟨ 1 , 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼
97	62, 4, 91, 10, 15, 2, 96	rloc1r 33454	. . . . 5 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ = (1r‘𝐿))
98	95, 97	sylan9eqr 2819	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1 ) → [⟨𝑥, 1 ⟩] ∼ = (1r‘𝐿))
99		fvexd 6882	. . . 4 ⊢ (𝜑 → (1r‘𝐿) ∈ V)
100	83, 98, 50, 99	fvmptd2 6984	. . 3 ⊢ (𝜑 → (𝐹‘ 1 ) = (1r‘𝐿))
101	30	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Ring)
102	50	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 ∈ 𝐵)
103	28, 29, 4, 101, 102	ringlidmd 20322	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ( 1 (.r‘𝑅) 1 ) = 1 )
104	103	eqcomd 2768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 = ( 1 (.r‘𝑅) 1 ))
105	104	opeq2d 4838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩)
106	105	eceq1d 8719	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
107	15	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ CRing)
108	2	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
109		simplr 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
110		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
111	108, 6	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 ∈ 𝑆)
112	28, 29, 90, 91, 10, 107, 108, 109, 110, 111, 111, 89	rlocmulval 33451	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ) = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
113	106, 112	eqtr4d 2800	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
114		opeq1 4831	. . . . . . 7 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩)
115	114	eceq1d 8719	. . . . . 6 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
116	28, 29, 101, 109, 110	ringcld 20310	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐵)
117		ecexg 8682	. . . . . . 7 ⊢ ( ∼ ∈ V → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
118	11, 117	mp1i 13	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
119	83, 115, 116, 118	fvmptd3 6999	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
120		opeq1 4831	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ⟨𝑥, 1 ⟩ = ⟨𝑎, 1 ⟩)
121	120	eceq1d 8719	. . . . . . 7 ⊢ (𝑥 = 𝑎 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑎, 1 ⟩] ∼ )
122		ecexg 8682	. . . . . . . 8 ⊢ ( ∼ ∈ V → [⟨𝑎, 1 ⟩] ∼ ∈ V)
123	11, 122	mp1i 13	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨𝑎, 1 ⟩] ∼ ∈ V)
124	83, 121, 109, 123	fvmptd3 6999	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑎) = [⟨𝑎, 1 ⟩] ∼ )
125		opeq1 4831	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ⟨𝑥, 1 ⟩ = ⟨𝑏, 1 ⟩)
126	125	eceq1d 8719	. . . . . . 7 ⊢ (𝑥 = 𝑏 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑏, 1 ⟩] ∼ )
127		ecexg 8682	. . . . . . . 8 ⊢ ( ∼ ∈ V → [⟨𝑏, 1 ⟩] ∼ ∈ V)
128	11, 127	mp1i 13	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨𝑏, 1 ⟩] ∼ ∈ V)
129	83, 126, 110, 128	fvmptd3 6999	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = [⟨𝑏, 1 ⟩] ∼ )
130	124, 129	oveq12d 7414	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
131	113, 119, 130	3eqtr4d 2807	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
132	131	anasss 470	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
133		eqid 2762	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
134		eqid 2762	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
135	13, 83	fmptd 7095	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ))
136	28, 62, 29, 57, 71, 91, 10, 15, 49	rlocbas 33449	. . . . 5 ⊢ (𝜑 → ((𝐵 × 𝑆) / ∼ ) = (Base‘𝐿))
137	136	feq3d 6676	. . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ) ↔ 𝐹:𝐵⟶(Base‘𝐿)))
138	135, 137	mpbid 234	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐿))
139	28, 29, 4, 101, 109	ringridmd 20323	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅) 1 ) = 𝑎)
140	28, 29, 4, 101, 110	ringridmd 20323	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(.r‘𝑅) 1 ) = 𝑏)
141	139, 140	oveq12d 7414	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )) = (𝑎(+g‘𝑅)𝑏))
142	141	eqcomd 2768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) = ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )))
143	142, 104	opeq12d 4839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩ = ⟨((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )), ( 1 (.r‘𝑅) 1 )⟩)
144	143	eceq1d 8719	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ = [⟨((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
145	28, 29, 90, 91, 10, 107, 108, 109, 110, 111, 111, 134	rlocaddval 33450	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ) = [⟨((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
146	144, 145	eqtr4d 2800	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
147		opeq1 4831	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩)
148	147	eceq1d 8719	. . . . . 6 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
149	16	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Grp)
150	28, 90, 149, 109, 110	grpcld 18989	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
151		ecexg 8682	. . . . . . 7 ⊢ ( ∼ ∈ V → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
152	11, 151	mp1i 13	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
153	83, 148, 150, 152	fvmptd3 6999	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
154	124, 129	oveq12d 7414	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
155	146, 153, 154	3eqtr4d 2807	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)))
156	155	anasss 470	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)))
157	28, 4, 88, 29, 89, 30, 93, 100, 132, 133, 90, 134, 138, 156	isrhmd 20537	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝐿))
158	87, 157	jca 519	1 ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ⊆ wss 3904  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   × cxp 5645  ⟶wf 6517  –1-1→wf1 6518  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  2^nd c2nd 7969  [cec 8676   / cqs 8677  Basecbs 17245  +_gcplusg 17286  ._rcmulr 17287  0_gc0g 17468  SubMndcsubmnd 18816  Grpcgrp 18975  -_gcsg 18977  mulGrpcmgp 20186  1_rcur 20231  Ringcrg 20283  CRingccrg 20284   RingHom crh 20518  RLRegcrlreg 20741   ~_RL cerl 33434   RLocal crloc 33435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-ec 8680  df-qs 8684  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-inf 9389  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-0g 17470  df-imas 17538  df-qus 17539  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-mhm 18817  df-submnd 18818  df-grp 18978  df-minusg 18979  df-sbg 18980  df-ghm 19254  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20232  df-ring 20285  df-cring 20286  df-rhm 20521  df-rlreg 20744  df-erl 33436  df-rloc 33437

This theorem is referenced by:  fracf1  33494