Theorem rlocf1 33354

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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
2		rlocf1.7	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
3		eqid 2739	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4		rlocf1.2	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
5	3, 4	ringidval 20155	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘𝑅))
6	5	subm0cl 18770	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
7	2, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝑆)
8	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ 𝑆)
9	1, 8	opelxpd 5657	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
10		rlocf1.4	. . . . . . 7 ⊢ ∼ = (𝑅 ~RL 𝑆)
11	10	ovexi 7390	. . . . . 6 ⊢ ∼ ∈ V
12	11	ecelqsi 8706	. . . . 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 3131	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
15		rlocf1.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing)
16	15	crnggrpd 20219	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
17	16	ad5antr 740	. . . . . . . 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 791	. . . . . . . 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 789	. . . . . . . 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 3435	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
21	4	fvexi 6841	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
22	20, 21	op1st 7939	. . . . . . . . . . . . 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 3435	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
25	24, 21	op2nd 7940	. . . . . . . . . . . . 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 7374	. . . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
30	15	crngringd 20218	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
31	30	ad5antr 740	. . . . . . . . . . . 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 20245	. . . . . . . . . . 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 2774	. . . . . . . . . 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 7939	. . . . . . . . . . . . 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 7940	. . . . . . . . . . . . 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 7374	. . . . . . . . . . 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 20245	. . . . . . . . . . 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 2774	. . . . . . . . . 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 7374	. . . . . . . . 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 740	. . . . . . . . . . 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 774	. . . . . . . . . . 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 3916	. . . . . . . . . 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 2839	. . . . . . . . . . . 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 20117	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
48	47	submss 18768	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
49	2, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
50	49, 7	sseldd 3916	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ 𝐵)
51	50	ad5antr 740	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 20232	. . . . . . . . . . 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 2839	. . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 20232	. . . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
58	28, 57	grpsubcl 18987	. . . . . . . . . . 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 1379	. . . . . . . . . 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 485	. . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
62		eqid 2739	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
63	61, 28, 29, 62	rrgeq0i 20671	. . . . . . . . . . 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 407	. . . . . . . . . 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 843	. . . . . . . . 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 2776	. . . . . . . 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 18993	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-g‘𝑅)𝑦) = (0g‘𝑅) ↔ 𝑥 = 𝑦))
68	67	biimpa 477	. . . . . . . 8 ⊢ (((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-g‘𝑅)𝑦) = (0g‘𝑅)) → 𝑥 = 𝑦)
69	17, 18, 19, 66, 68	syl31anc 1381	. . . . . . 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 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑆 ⊆ 𝐵)
71		eqid 2739	. . . . . . . . . . 11 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
72	15	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
73	2	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
74	28, 62, 4, 29, 57, 71, 10, 72, 73	erler 33346	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ∼ Er (𝐵 × 𝑆))
75	9	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
76	74, 75	erth 8688	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩ ↔ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ))
77	76	biimpar 478	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → ⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩)
78	28, 10, 70, 62, 29, 57, 77	erldi 33343	. . . . . . 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 3147	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑥 = 𝑦)
80	79	ex 413	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
81	80	anasss 467	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
82	81	ralrimivva 3182	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
83		rlocf1.5	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )
84		opeq1 4804	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 1 ⟩ = ⟨𝑦, 1 ⟩)
85	84	eceq1d 8674	. . . 4 ⊢ (𝑥 = 𝑦 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ )
86	83, 85	f1mpt 7205	. . 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 589	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ))
88		eqid 2739	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
89		eqid 2739	. . 3 ⊢ (.r‘𝐿) = (.r‘𝐿)
90		eqid 2739	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
91		rlocf1.3	. . . . 5 ⊢ 𝐿 = (𝑅 RLocal 𝑆)
92	28, 29, 90, 91, 10, 15, 2	rloccring 33351	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
93	92	crngringd 20218	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
94		opeq1 4804	. . . . . 6 ⊢ (𝑥 = 1 → ⟨𝑥, 1 ⟩ = ⟨ 1 , 1 ⟩)
95	94	eceq1d 8674	. . . . 5 ⊢ (𝑥 = 1 → [⟨𝑥, 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼ )
96		eqid 2739	. . . . . 6 ⊢ [⟨ 1 , 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼
97	62, 4, 91, 10, 15, 2, 96	rloc1r 33353	. . . . 5 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ = (1r‘𝐿))
98	95, 97	sylan9eqr 2796	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1 ) → [⟨𝑥, 1 ⟩] ∼ = (1r‘𝐿))
99		fvexd 6842	. . . 4 ⊢ (𝜑 → (1r‘𝐿) ∈ V)
100	83, 98, 50, 99	fvmptd2 6944	. . 3 ⊢ (𝜑 → (𝐹‘ 1 ) = (1r‘𝐿))
101	30	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Ring)
102	50	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 ∈ 𝐵)
103	28, 29, 4, 101, 102	ringlidmd 20244	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ( 1 (.r‘𝑅) 1 ) = 1 )
104	103	eqcomd 2745	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 = ( 1 (.r‘𝑅) 1 ))
105	104	opeq2d 4811	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩)
106	105	eceq1d 8674	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
107	15	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ CRing)
108	2	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
109		simplr 774	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
110		simpr 485	. . . . . . 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 33350	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ) = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
113	106, 112	eqtr4d 2777	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
114		opeq1 4804	. . . . . . 7 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩)
115	114	eceq1d 8674	. . . . . 6 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
116	28, 29, 101, 109, 110	ringcld 20232	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐵)
117		ecexg 8637	. . . . . . 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 6959	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
120		opeq1 4804	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ⟨𝑥, 1 ⟩ = ⟨𝑎, 1 ⟩)
121	120	eceq1d 8674	. . . . . . 7 ⊢ (𝑥 = 𝑎 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑎, 1 ⟩] ∼ )
122		ecexg 8637	. . . . . . . 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 6959	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑎) = [⟨𝑎, 1 ⟩] ∼ )
125		opeq1 4804	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ⟨𝑥, 1 ⟩ = ⟨𝑏, 1 ⟩)
126	125	eceq1d 8674	. . . . . . 7 ⊢ (𝑥 = 𝑏 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑏, 1 ⟩] ∼ )
127		ecexg 8637	. . . . . . . 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 6959	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = [⟨𝑏, 1 ⟩] ∼ )
130	124, 129	oveq12d 7374	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
131	113, 119, 130	3eqtr4d 2784	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
132	131	anasss 467	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
133		eqid 2739	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
134		eqid 2739	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
135	13, 83	fmptd 7055	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ))
136	28, 62, 29, 57, 71, 91, 10, 15, 49	rlocbas 33348	. . . . 5 ⊢ (𝜑 → ((𝐵 × 𝑆) / ∼ ) = (Base‘𝐿))
137	136	feq3d 6640	. . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ) ↔ 𝐹:𝐵⟶(Base‘𝐿)))
138	135, 137	mpbid 233	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐿))
139	28, 29, 4, 101, 109	ringridmd 20245	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅) 1 ) = 𝑎)
140	28, 29, 4, 101, 110	ringridmd 20245	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(.r‘𝑅) 1 ) = 𝑏)
141	139, 140	oveq12d 7374	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )) = (𝑎(+g‘𝑅)𝑏))
142	141	eqcomd 2745	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) = ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )))
143	142, 104	opeq12d 4812	. . . . . . 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 8674	. . . . . 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 33349	. . . . . 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 2777	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
147		opeq1 4804	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩)
148	147	eceq1d 8674	. . . . . 6 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
149	16	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Grp)
150	28, 90, 149, 109, 110	grpcld 18914	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
151		ecexg 8637	. . . . . . 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 6959	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
154	124, 129	oveq12d 7374	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
155	146, 153, 154	3eqtr4d 2784	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)))
156	155	anasss 467	. . 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 20459	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝐿))
158	87, 157	jca 516	1 ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883  ⟨cop 4561   class class class wbr 5072   ↦ cmpt 5153   × cxp 5616  ⟶wf 6481  –1-1→wf1 6482  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  [cec 8631   / cqs 8632  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212  0_gc0g 17393  SubMndcsubmnd 18741  Grpcgrp 18900  -_gcsg 18902  mulGrpcmgp 20112  1_rcur 20153  Ringcrg 20205  CRingccrg 20206   RingHom crh 20440  RLRegcrlreg 20663   ~_RL cerl 33334   RLocal crloc 33335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-ec 8635  df-qs 8639  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-0g 17395  df-imas 17463  df-qus 17464  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-ghm 19179  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-rhm 20443  df-rlreg 20666  df-erl 33336  df-rloc 33337

This theorem is referenced by:  fracf1  33391