Theorem rlocf1 33278

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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
2		rlocf1.7	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
3		eqid 2736	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4		rlocf1.2	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
5	3, 4	ringidval 20181	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘𝑅))
6	5	subm0cl 18825	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
7	2, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝑆)
8	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ 𝑆)
9	1, 8	opelxpd 5723	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
10		rlocf1.4	. . . . . . 7 ⊢ ∼ = (𝑅 ~RL 𝑆)
11	10	ovexi 7466	. . . . . 6 ⊢ ∼ ∈ V
12	11	ecelqsi 8814	. . . . 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 3145	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
15		rlocf1.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing)
16	15	crnggrpd 20245	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
17	16	ad5antr 734	. . . . . . . 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 785	. . . . . . . 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 783	. . . . . . . 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 3483	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
21	4	fvexi 6919	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
22	20, 21	op1st 8023	. . . . . . . . . . . . 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 3483	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
25	24, 21	op2nd 8024	. . . . . . . . . . . . 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 7450	. . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
30	15	crngringd 20244	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
31	30	ad5antr 734	. . . . . . . . . . . 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 20271	. . . . . . . . . . 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 2776	. . . . . . . . . 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 8023	. . . . . . . . . . . . 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 8024	. . . . . . . . . . . . 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 7450	. . . . . . . . . . 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 20271	. . . . . . . . . . 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 2776	. . . . . . . . . 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 7450	. . . . . . . . 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 734	. . . . . . . . . . 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 768	. . . . . . . . . . 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 3983	. . . . . . . . . 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 2840	. . . . . . . . . . . 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 20143	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
48	47	submss 18823	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
49	2, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
50	49, 7	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ 𝐵)
51	50	ad5antr 734	. . . . . . . . . . . . 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 2840	. . . . . . . . . . . 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 20258	. . . . . . . . . . 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 2840	. . . . . . . . . . . 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 2840	. . . . . . . . . . . 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 20258	. . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
58	28, 57	grpsubcl 19039	. . . . . . . . . . 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 1372	. . . . . . . . . 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 484	. . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
62		eqid 2736	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
63	61, 28, 29, 62	rrgeq0i 20700	. . . . . . . . . . 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 406	. . . . . . . . . 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 837	. . . . . . . . 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 2778	. . . . . . . 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 19045	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-g‘𝑅)𝑦) = (0g‘𝑅) ↔ 𝑥 = 𝑦))
68	67	biimpa 476	. . . . . . . 8 ⊢ (((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-g‘𝑅)𝑦) = (0g‘𝑅)) → 𝑥 = 𝑦)
69	17, 18, 19, 66, 68	syl31anc 1374	. . . . . . 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 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑆 ⊆ 𝐵)
71		eqid 2736	. . . . . . . . . . 11 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
72	15	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
73	2	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
74	28, 62, 4, 29, 57, 71, 10, 72, 73	erler 33270	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ∼ Er (𝐵 × 𝑆))
75	9	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
76	74, 75	erth 8797	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩ ↔ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ))
77	76	biimpar 477	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → ⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩)
78	28, 10, 70, 62, 29, 57, 77	erldi 33267	. . . . . . 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 3161	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑥 = 𝑦)
80	79	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
81	80	anasss 466	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
82	81	ralrimivva 3201	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
83		rlocf1.5	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )
84		opeq1 4872	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 1 ⟩ = ⟨𝑦, 1 ⟩)
85	84	eceq1d 8786	. . . 4 ⊢ (𝑥 = 𝑦 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ )
86	83, 85	f1mpt 7282	. . 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 583	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ))
88		eqid 2736	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
89		eqid 2736	. . 3 ⊢ (.r‘𝐿) = (.r‘𝐿)
90		eqid 2736	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
91		rlocf1.3	. . . . 5 ⊢ 𝐿 = (𝑅 RLocal 𝑆)
92	28, 29, 90, 91, 10, 15, 2	rloccring 33275	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
93	92	crngringd 20244	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
94		opeq1 4872	. . . . . 6 ⊢ (𝑥 = 1 → ⟨𝑥, 1 ⟩ = ⟨ 1 , 1 ⟩)
95	94	eceq1d 8786	. . . . 5 ⊢ (𝑥 = 1 → [⟨𝑥, 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼ )
96		eqid 2736	. . . . . 6 ⊢ [⟨ 1 , 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼
97	62, 4, 91, 10, 15, 2, 96	rloc1r 33277	. . . . 5 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ = (1r‘𝐿))
98	95, 97	sylan9eqr 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1 ) → [⟨𝑥, 1 ⟩] ∼ = (1r‘𝐿))
99		fvexd 6920	. . . 4 ⊢ (𝜑 → (1r‘𝐿) ∈ V)
100	83, 98, 50, 99	fvmptd2 7023	. . 3 ⊢ (𝜑 → (𝐹‘ 1 ) = (1r‘𝐿))
101	30	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Ring)
102	50	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 ∈ 𝐵)
103	28, 29, 4, 101, 102	ringlidmd 20270	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ( 1 (.r‘𝑅) 1 ) = 1 )
104	103	eqcomd 2742	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 = ( 1 (.r‘𝑅) 1 ))
105	104	opeq2d 4879	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩)
106	105	eceq1d 8786	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
107	15	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ CRing)
108	2	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
109		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
110		simpr 484	. . . . . . 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 33274	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ) = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
113	106, 112	eqtr4d 2779	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
114		opeq1 4872	. . . . . . 7 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩)
115	114	eceq1d 8786	. . . . . 6 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
116	28, 29, 101, 109, 110	ringcld 20258	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐵)
117		ecexg 8750	. . . . . . 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 7038	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
120		opeq1 4872	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ⟨𝑥, 1 ⟩ = ⟨𝑎, 1 ⟩)
121	120	eceq1d 8786	. . . . . . 7 ⊢ (𝑥 = 𝑎 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑎, 1 ⟩] ∼ )
122		ecexg 8750	. . . . . . . 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 7038	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑎) = [⟨𝑎, 1 ⟩] ∼ )
125		opeq1 4872	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ⟨𝑥, 1 ⟩ = ⟨𝑏, 1 ⟩)
126	125	eceq1d 8786	. . . . . . 7 ⊢ (𝑥 = 𝑏 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑏, 1 ⟩] ∼ )
127		ecexg 8750	. . . . . . . 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 7038	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = [⟨𝑏, 1 ⟩] ∼ )
130	124, 129	oveq12d 7450	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
131	113, 119, 130	3eqtr4d 2786	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
132	131	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
133		eqid 2736	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
134		eqid 2736	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
135	13, 83	fmptd 7133	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ))
136	28, 62, 29, 57, 71, 91, 10, 15, 49	rlocbas 33272	. . . . 5 ⊢ (𝜑 → ((𝐵 × 𝑆) / ∼ ) = (Base‘𝐿))
137	136	feq3d 6722	. . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ) ↔ 𝐹:𝐵⟶(Base‘𝐿)))
138	135, 137	mpbid 232	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐿))
139	28, 29, 4, 101, 109	ringridmd 20271	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅) 1 ) = 𝑎)
140	28, 29, 4, 101, 110	ringridmd 20271	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(.r‘𝑅) 1 ) = 𝑏)
141	139, 140	oveq12d 7450	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )) = (𝑎(+g‘𝑅)𝑏))
142	141	eqcomd 2742	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) = ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )))
143	142, 104	opeq12d 4880	. . . . . . 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 8786	. . . . . 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 33273	. . . . . 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 2779	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
147		opeq1 4872	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩)
148	147	eceq1d 8786	. . . . . 6 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
149	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Grp)
150	28, 90, 149, 109, 110	grpcld 18966	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
151		ecexg 8750	. . . . . . 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 7038	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
154	124, 129	oveq12d 7450	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
155	146, 153, 154	3eqtr4d 2786	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)))
156	155	anasss 466	. . 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 20489	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝐿))
158	87, 157	jca 511	1 ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ⊆ wss 3950  ⟨cop 4631   class class class wbr 5142   ↦ cmpt 5224   × cxp 5682  ⟶wf 6556  –1-1→wf1 6557  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  [cec 8744   / cqs 8745  Basecbs 17248  +_gcplusg 17298  ._rcmulr 17299  0_gc0g 17485  SubMndcsubmnd 18796  Grpcgrp 18952  -_gcsg 18954  mulGrpcmgp 20138  1_rcur 20179  Ringcrg 20231  CRingccrg 20232   RingHom crh 20470  RLRegcrlreg 20692   ~_RL cerl 33258   RLocal crloc 33259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-ec 8748  df-qs 8752  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17487  df-imas 17554  df-qus 17555  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-ghm 19232  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-rhm 20473  df-rlreg 20695  df-erl 33260  df-rloc 33261

This theorem is referenced by:  fracf1  33310