Theorem rlocf1 33835

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 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
2		rlocf1.7	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
3		eqid 2761	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4		rlocf1.2	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
5	3, 4	ringidval 20409	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘𝑅))
6	5	subm0cl 19006	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
7	2, 6	syl 18	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝑆)
8	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ 𝑆)
9	1, 8	opelxpd 5690	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
10		rlocf1.4	. . . . . . 7 ⊢ ∼ = (𝑅 ~RL 𝑆)
11	10	ovexi 7454	. . . . . 6 ⊢ ∼ ∈ V
12	11	ecelqsi 8790	. . . . 5 ⊢ (⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆) → [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
13	9, 12	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
14	13	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 [⟨𝑥, 1 ⟩] ∼ ∈ ((𝐵 × 𝑆) / ∼ ))
15		rlocf1.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing)
16	15	crnggrpd 20474	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
17	16	ad5antr 747	. . . . . . . 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 798	. . . . . . . 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 796	. . . . . . . 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 3455	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
21	4	fvexi 6899	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
22	20, 21	op1st 8009	. . . . . . . . . . . . 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 3455	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
25	24, 21	op2nd 8010	. . . . . . . . . . . . 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 7438	. . . . . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
30	15	crngringd 20473	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
31	30	ad5antr 747	. . . . . . . . . . . 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 20502	. . . . . . . . . . 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 2796	. . . . . . . . . 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 8009	. . . . . . . . . . . . 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 8010	. . . . . . . . . . . . 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 7438	. . . . . . . . . . 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 20502	. . . . . . . . . . 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 2796	. . . . . . . . . 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 7438	. . . . . . . . 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 747	. . . . . . . . . . 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 781	. . . . . . . . . . 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 3932	. . . . . . . . . 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 2861	. . . . . . . . . . . 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 20365	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
48	47	submss 19004	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
49	2, 48	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
50	49, 7	sseldd 3932	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ 𝐵)
51	50	ad5antr 747	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . 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 20484	. . . . . . . . . . 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 2861	. . . . . . . . . . . 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 2861	. . . . . . . . . . . 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 20484	. . . . . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
58	28, 57	grpsubcl 19230	. . . . . . . . . . 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 1398	. . . . . . . . . 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 490	. . . . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
62		eqid 2761	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
63	61, 28, 29, 62	rrgeq0i 20951	. . . . . . . . . . 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 412	. . . . . . . . . 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 851	. . . . . . . . 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 2798	. . . . . . . 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 19236	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-g‘𝑅)𝑦) = (0g‘𝑅) ↔ 𝑥 = 𝑦))
68	67	biimpa 482	. . . . . . . 8 ⊢ (((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-g‘𝑅)𝑦) = (0g‘𝑅)) → 𝑥 = 𝑦)
69	17, 18, 19, 66, 68	syl31anc 1400	. . . . . . 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 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑆 ⊆ 𝐵)
71		eqid 2761	. . . . . . . . . . 11 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
72	15	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
73	2	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
74	28, 62, 4, 29, 57, 71, 10, 72, 73	erler 33826	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ∼ Er (𝐵 × 𝑆))
75	9	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 1 ⟩ ∈ (𝐵 × 𝑆))
76	74, 75	erth 8772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩ ↔ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ))
77	76	biimpar 483	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → ⟨𝑥, 1 ⟩ ∼ ⟨𝑦, 1 ⟩)
78	28, 10, 70, 62, 29, 57, 77	erldi 33823	. . . . . . 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 3171	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ ) → 𝑥 = 𝑦)
80	79	ex 418	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
81	80	anasss 472	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
82	81	ralrimivva 3206	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ([⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ → 𝑥 = 𝑦))
83		rlocf1.5	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )
84		opeq1 4833	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 1 ⟩ = ⟨𝑦, 1 ⟩)
85	84	eceq1d 8758	. . . 4 ⊢ (𝑥 = 𝑦 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑦, 1 ⟩] ∼ )
86	83, 85	f1mpt 7265	. . 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 595	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ))
88		eqid 2761	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
89		eqid 2761	. . 3 ⊢ (.r‘𝐿) = (.r‘𝐿)
90		eqid 2761	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
91		rlocf1.3	. . . . 5 ⊢ 𝐿 = (𝑅 RLocal 𝑆)
92	28, 29, 90, 91, 10, 15, 2	rloccring 33832	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
93	92	crngringd 20473	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
94		opeq1 4833	. . . . . 6 ⊢ (𝑥 = 1 → ⟨𝑥, 1 ⟩ = ⟨ 1 , 1 ⟩)
95	94	eceq1d 8758	. . . . 5 ⊢ (𝑥 = 1 → [⟨𝑥, 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼ )
96		eqid 2761	. . . . . 6 ⊢ [⟨ 1 , 1 ⟩] ∼ = [⟨ 1 , 1 ⟩] ∼
97	62, 4, 91, 10, 15, 2, 96	rloc1r 33834	. . . . 5 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ = (1r‘𝐿))
98	95, 97	sylan9eqr 2818	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 1 ) → [⟨𝑥, 1 ⟩] ∼ = (1r‘𝐿))
99		fvexd 6900	. . . 4 ⊢ (𝜑 → (1r‘𝐿) ∈ V)
100	83, 98, 50, 99	fvmptd2 7002	. . 3 ⊢ (𝜑 → (𝐹‘ 1 ) = (1r‘𝐿))
101	30	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Ring)
102	50	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 ∈ 𝐵)
103	28, 29, 4, 101, 102	ringlidmd 20501	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ( 1 (.r‘𝑅) 1 ) = 1 )
104	103	eqcomd 2767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 = ( 1 (.r‘𝑅) 1 ))
105	104	opeq2d 4840	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩)
106	105	eceq1d 8758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
107	15	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ CRing)
108	2	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
109		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
110		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
111	108, 6	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 1 ∈ 𝑆)
112	28, 29, 90, 91, 10, 107, 108, 109, 110, 111, 111, 89	rlocmulval 33831	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ) = [⟨(𝑎(.r‘𝑅)𝑏), ( 1 (.r‘𝑅) 1 )⟩] ∼ )
113	106, 112	eqtr4d 2799	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
114		opeq1 4833	. . . . . . 7 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩)
115	114	eceq1d 8758	. . . . . 6 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
116	28, 29, 101, 109, 110	ringcld 20484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐵)
117		ecexg 8721	. . . . . . 7 ⊢ ( ∼ ∈ V → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
118	11, 117	mp1i 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
119	83, 115, 116, 118	fvmptd3 7017	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = [⟨(𝑎(.r‘𝑅)𝑏), 1 ⟩] ∼ )
120		opeq1 4833	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ⟨𝑥, 1 ⟩ = ⟨𝑎, 1 ⟩)
121	120	eceq1d 8758	. . . . . . 7 ⊢ (𝑥 = 𝑎 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑎, 1 ⟩] ∼ )
122		ecexg 8721	. . . . . . . 8 ⊢ ( ∼ ∈ V → [⟨𝑎, 1 ⟩] ∼ ∈ V)
123	11, 122	mp1i 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨𝑎, 1 ⟩] ∼ ∈ V)
124	83, 121, 109, 123	fvmptd3 7017	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑎) = [⟨𝑎, 1 ⟩] ∼ )
125		opeq1 4833	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ⟨𝑥, 1 ⟩ = ⟨𝑏, 1 ⟩)
126	125	eceq1d 8758	. . . . . . 7 ⊢ (𝑥 = 𝑏 → [⟨𝑥, 1 ⟩] ∼ = [⟨𝑏, 1 ⟩] ∼ )
127		ecexg 8721	. . . . . . . 8 ⊢ ( ∼ ∈ V → [⟨𝑏, 1 ⟩] ∼ ∈ V)
128	11, 127	mp1i 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨𝑏, 1 ⟩] ∼ ∈ V)
129	83, 126, 110, 128	fvmptd3 7017	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = [⟨𝑏, 1 ⟩] ∼ )
130	124, 129	oveq12d 7438	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (.r‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
131	113, 119, 130	3eqtr4d 2806	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
132	131	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝐿)(𝐹‘𝑏)))
133		eqid 2761	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
134		eqid 2761	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
135	13, 83	fmptd 7114	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ))
136	28, 62, 29, 57, 71, 91, 10, 15, 49	rlocbas 33829	. . . . 5 ⊢ (𝜑 → ((𝐵 × 𝑆) / ∼ ) = (Base‘𝐿))
137	136	feq3d 6694	. . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶((𝐵 × 𝑆) / ∼ ) ↔ 𝐹:𝐵⟶(Base‘𝐿)))
138	135, 137	mpbid 235	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐿))
139	28, 29, 4, 101, 109	ringridmd 20502	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(.r‘𝑅) 1 ) = 𝑎)
140	28, 29, 4, 101, 110	ringridmd 20502	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(.r‘𝑅) 1 ) = 𝑏)
141	139, 140	oveq12d 7438	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )) = (𝑎(+g‘𝑅)𝑏))
142	141	eqcomd 2767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) = ((𝑎(.r‘𝑅) 1 )(+g‘𝑅)(𝑏(.r‘𝑅) 1 )))
143	142, 104	opeq12d 4841	. . . . . . 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 8758	. . . . . 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 33830	. . . . . 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 2799	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
147		opeq1 4833	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → ⟨𝑥, 1 ⟩ = ⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩)
148	147	eceq1d 8758	. . . . . 6 ⊢ (𝑥 = (𝑎(+g‘𝑅)𝑏) → [⟨𝑥, 1 ⟩] ∼ = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
149	16	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → 𝑅 ∈ Grp)
150	28, 90, 149, 109, 110	grpcld 19158	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
151		ecexg 8721	. . . . . . 7 ⊢ ( ∼ ∈ V → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
152	11, 151	mp1i 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ ∈ V)
153	83, 148, 150, 152	fvmptd3 7017	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = [⟨(𝑎(+g‘𝑅)𝑏), 1 ⟩] ∼ )
154	124, 129	oveq12d 7438	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)) = ([⟨𝑎, 1 ⟩] ∼ (+g‘𝐿)[⟨𝑏, 1 ⟩] ∼ ))
155	146, 153, 154	3eqtr4d 2806	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝐿)(𝐹‘𝑏)))
156	155	anasss 472	. . 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 20722	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝐿))
158	87, 157	jca 521	1 ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  [cec 8715   / cqs 8716  Basecbs 17387  +_gcplusg 17428  ._rcmulr 17429  0_gc0g 17610  SubMndcsubmnd 18977  Grpcgrp 19144  -_gcsg 19146  mulGrpcmgp 20360  1_rcur 20407  Ringcrg 20459  CRingccrg 20460   RingHom crh 20699  RLRegcrlreg 20943   ~_RL cerl 33814   RLocal crloc 33815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  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-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-ec 8719  df-qs 8723  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-0g 17612  df-imas 17680  df-qus 17681  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-ghm 19428  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-rhm 20702  df-rlreg 20946  df-erl 33816  df-rloc 33817

This theorem is used by:  fracf1  33869