Theorem fracfld 33275

Description: The field of fractions of an integral domain is a field. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypothesis

Ref	Expression
fracfld.1	⊢ (𝜑 → 𝑅 ∈ IDomn)

Assertion

Ref	Expression
fracfld	⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)

Proof of Theorem fracfld

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

Step	Hyp	Ref	Expression
1		fracval 33271	. 2 ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
2		fracfld.1	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ IDomn)
3	2	idomdomd 20748	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Domn)
4		domnnzr 20728	. . . . . . 7 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
5		eqid 2740	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
6		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
7	5, 6	nzrnz 20541	. . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅))
8	3, 4, 7	3syl 18	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))
9		fvex 6933	. . . . . . . . . . . . . . . . 17 ⊢ (1r‘𝑅) ∈ V
10	9, 9	op1st 8038	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩) = (1r‘𝑅)
11	10	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩) = (1r‘𝑅))
12		fvex 6933	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) ∈ V
13	12, 9	op2nd 8039	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩) = (1r‘𝑅)
14	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩) = (1r‘𝑅))
15	11, 14	oveq12d 7466	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)) = ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)))
16		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑅) = (.r‘𝑅)
18	2	idomringd 20750	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
19	18	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑅 ∈ Ring)
20	16, 5	ringidcl 20289	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (1r‘𝑅) ∈ (Base‘𝑅))
22	16, 17, 5, 19, 21	ringlidmd 20295	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
23	15, 22	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)) = (1r‘𝑅))
24	12, 9	op1st 8038	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩) = (0g‘𝑅)
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩) = (0g‘𝑅))
26	9, 9	op2nd 8039	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩) = (1r‘𝑅)
27	26	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩) = (1r‘𝑅))
28	25, 27	oveq12d 7466	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)) = ((0g‘𝑅)(.r‘𝑅)(1r‘𝑅)))
29	18	ringgrpd 20269	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ Grp)
30	16, 6	grpidcl 19005	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ (Base‘𝑅))
31	29, 30	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
32	31	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (0g‘𝑅) ∈ (Base‘𝑅))
33	16, 17, 5, 19, 32	ringridmd 20296	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (0g‘𝑅))
34	28, 33	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)) = (0g‘𝑅))
35	23, 34	oveq12d 7466	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩))) = ((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)))
36	35	oveq2d 7464	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r‘𝑅)(((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)))) = (𝑡(.r‘𝑅)((1r‘𝑅)(-g‘𝑅)(0g‘𝑅))))
37		eqid 2740	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
38		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
39	38, 16	rrgss 20724	. . . . . . . . . . . . . 14 ⊢ (RLReg‘𝑅) ⊆ (Base‘𝑅)
40	39	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (RLReg‘𝑅) ⊆ (Base‘𝑅))
41	40	sselda 4008	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑡 ∈ (Base‘𝑅))
42	16, 17, 37, 19, 41, 21, 32	ringsubdi 20330	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r‘𝑅)((1r‘𝑅)(-g‘𝑅)(0g‘𝑅))) = ((𝑡(.r‘𝑅)(1r‘𝑅))(-g‘𝑅)(𝑡(.r‘𝑅)(0g‘𝑅))))
43	16, 17, 5, 19, 41	ringridmd 20296	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r‘𝑅)(1r‘𝑅)) = 𝑡)
44	16, 17, 6, 19, 41	ringrzd 20319	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
45	43, 44	oveq12d 7466	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((𝑡(.r‘𝑅)(1r‘𝑅))(-g‘𝑅)(𝑡(.r‘𝑅)(0g‘𝑅))) = (𝑡(-g‘𝑅)(0g‘𝑅)))
46	29	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑅 ∈ Grp)
47	16, 6, 37	grpsubid1 19065	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑡 ∈ (Base‘𝑅)) → (𝑡(-g‘𝑅)(0g‘𝑅)) = 𝑡)
48	46, 41, 47	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(-g‘𝑅)(0g‘𝑅)) = 𝑡)
49	45, 48	eqtrd 2780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((𝑡(.r‘𝑅)(1r‘𝑅))(-g‘𝑅)(𝑡(.r‘𝑅)(0g‘𝑅))) = 𝑡)
50	36, 42, 49	3eqtrd 2784	. . . . . . . . . 10 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r‘𝑅)(((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)))) = 𝑡)
51	50	eqeq1d 2742	. . . . . . . . 9 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((𝑡(.r‘𝑅)(((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)))) = (0g‘𝑅) ↔ 𝑡 = (0g‘𝑅)))
52	51	biimpa 476	. . . . . . . 8 ⊢ ((((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)))) = (0g‘𝑅)) → 𝑡 = (0g‘𝑅))
53		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑡 ∈ (RLReg‘𝑅))
54	38, 6	rrgnz 20726	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ NzRing → ¬ (0g‘𝑅) ∈ (RLReg‘𝑅))
55	3, 4, 54	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (0g‘𝑅) ∈ (RLReg‘𝑅))
56	55	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ¬ (0g‘𝑅) ∈ (RLReg‘𝑅))
57		nelne2 3046	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (RLReg‘𝑅) ∧ ¬ (0g‘𝑅) ∈ (RLReg‘𝑅)) → 𝑡 ≠ (0g‘𝑅))
58	53, 56, 57	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑡 ≠ (0g‘𝑅))
59	58	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)))) = (0g‘𝑅)) → 𝑡 ≠ (0g‘𝑅))
60	52, 59	pm2.21ddne 3032	. . . . . . 7 ⊢ ((((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡(.r‘𝑅)(((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)))) = (0g‘𝑅)) → (1r‘𝑅) = (0g‘𝑅))
61		eqid 2740	. . . . . . . 8 ⊢ (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
62		eqid 2740	. . . . . . . . . . 11 ⊢ ((Base‘𝑅) × (RLReg‘𝑅)) = ((Base‘𝑅) × (RLReg‘𝑅))
63	2	idomcringd 20749	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ CRing)
64	16, 38, 6	isdomn6 20736	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅)))
65	3, 64	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ NzRing ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅)))
66	65	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅))
67		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
68	16, 6, 67	isdomn3 20737	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ Ring ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅))))
69	3, 68	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅))))
70	69	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅)))
71	66, 70	eqeltrrd 2845	. . . . . . . . . . 11 ⊢ (𝜑 → (RLReg‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅)))
72	16, 6, 5, 17, 37, 62, 61, 63, 71	erler 33237	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ~RL (RLReg‘𝑅)) Er ((Base‘𝑅) × (RLReg‘𝑅)))
73	18, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
74	5, 38, 18	1rrg 33252	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝑅) ∈ (RLReg‘𝑅))
75	73, 74	opelxpd 5739	. . . . . . . . . 10 ⊢ (𝜑 → ⟨(1r‘𝑅), (1r‘𝑅)⟩ ∈ ((Base‘𝑅) × (RLReg‘𝑅)))
76	72, 75	erth 8814	. . . . . . . . 9 ⊢ (𝜑 → (⟨(1r‘𝑅), (1r‘𝑅)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g‘𝑅), (1r‘𝑅)⟩ ↔ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
77	76	biimpar 477	. . . . . . . 8 ⊢ ((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → ⟨(1r‘𝑅), (1r‘𝑅)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g‘𝑅), (1r‘𝑅)⟩)
78	16, 61, 40, 6, 17, 37, 77	erldi 33234	. . . . . . 7 ⊢ ((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → ∃𝑡 ∈ (RLReg‘𝑅)(𝑡(.r‘𝑅)(((1st ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(0g‘𝑅), (1r‘𝑅)⟩))(-g‘𝑅)((1st ‘⟨(0g‘𝑅), (1r‘𝑅)⟩)(.r‘𝑅)(2nd ‘⟨(1r‘𝑅), (1r‘𝑅)⟩)))) = (0g‘𝑅))
79	60, 78	r19.29a 3168	. . . . . 6 ⊢ ((𝜑 ∧ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (1r‘𝑅) = (0g‘𝑅))
80	8, 79	mteqand 3039	. . . . 5 ⊢ (𝜑 → [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) ≠ [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
81		eqid 2740	. . . . . 6 ⊢ (𝑅 RLocal (RLReg‘𝑅)) = (𝑅 RLocal (RLReg‘𝑅))
82		eqid 2740	. . . . . 6 ⊢ [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))
83	6, 5, 81, 61, 63, 71, 82	rloc1r 33244	. . . . 5 ⊢ (𝜑 → [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
84		eqid 2740	. . . . . 6 ⊢ [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))
85	6, 5, 81, 61, 63, 71, 84	rloc0g 33243	. . . . 5 ⊢ (𝜑 → [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
86	80, 83, 85	3netr3d 3023	. . . 4 ⊢ (𝜑 → (1r‘(𝑅 RLocal (RLReg‘𝑅))) ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
87		oveq2 7456	. . . . . . . . 9 ⊢ (𝑦 = [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) → (𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑦) = (𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))))
88	87	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑦 = [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) → ((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑦) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ↔ (𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))))
89		oveq1 7455	. . . . . . . . 9 ⊢ (𝑦 = [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) → (𝑦(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥))
90	89	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑦 = [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) → ((𝑦(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ↔ ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))))
91	88, 90	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) → (((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑦) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ∧ (𝑦(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))) ↔ ((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ∧ ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))))
92		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑏 ∈ (RLReg‘𝑅))
93	39, 92	sselid 4006	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑏 ∈ (Base‘𝑅))
94		simpllr 775	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑎 ∈ (Base‘𝑅))
95		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅)))
96	72	ad5antr 733	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → (𝑅 ~RL (RLReg‘𝑅)) Er ((Base‘𝑅) × (RLReg‘𝑅)))
97	18	ad5antr 733	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑅 ∈ Ring)
98	97, 20	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → (1r‘𝑅) ∈ (Base‘𝑅))
99	16, 17, 6, 97, 98	ringlzd 20318	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (0g‘𝑅))
100		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑎 = (0g‘𝑅))
101	100	oveq1d 7463	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → (𝑎(.r‘𝑅)(1r‘𝑅)) = ((0g‘𝑅)(.r‘𝑅)(1r‘𝑅)))
102	93	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑏 ∈ (Base‘𝑅))
103	16, 17, 6, 97, 102	ringlzd 20318	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑏) = (0g‘𝑅))
104	99, 101, 103	3eqtr4d 2790	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → (𝑎(.r‘𝑅)(1r‘𝑅)) = ((0g‘𝑅)(.r‘𝑅)𝑏))
105	63	ad5antr 733	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑅 ∈ CRing)
106	94	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑎 ∈ (Base‘𝑅))
107	31	ad5antr 733	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → (0g‘𝑅) ∈ (Base‘𝑅))
108	92	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑏 ∈ (RLReg‘𝑅))
109	74	ad5antr 733	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → (1r‘𝑅) ∈ (RLReg‘𝑅))
110	16, 17, 61, 105, 106, 107, 108, 109	fracerl 33273	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → (⟨𝑎, 𝑏⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g‘𝑅), (1r‘𝑅)⟩ ↔ (𝑎(.r‘𝑅)(1r‘𝑅)) = ((0g‘𝑅)(.r‘𝑅)𝑏)))
111	104, 110	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → ⟨𝑎, 𝑏⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g‘𝑅), (1r‘𝑅)⟩)
112	96, 111	erthi 8816	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
113	85	ad5antr 733	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
114	95, 112, 113	3eqtrd 2784	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑥 = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
115		eldifsni 4815	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}) → 𝑥 ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
116	115	ad5antlr 734	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → 𝑥 ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
117	116	neneqd 2951	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g‘𝑅)) → ¬ 𝑥 = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
118	114, 117	pm2.65da 816	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ¬ 𝑎 = (0g‘𝑅))
119	118	neqned 2953	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑎 ≠ (0g‘𝑅))
120	94, 119	eldifsnd 4812	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑎 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
121	66	ad4antr 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅))
122	120, 121	eleqtrd 2846	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑎 ∈ (RLReg‘𝑅))
123	93, 122	opelxpd 5739	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ⟨𝑏, 𝑎⟩ ∈ ((Base‘𝑅) × (RLReg‘𝑅)))
124		ovex 7481	. . . . . . . . . 10 ⊢ (𝑅 ~RL (RLReg‘𝑅)) ∈ V
125	124	ecelqsi 8831	. . . . . . . . 9 ⊢ (⟨𝑏, 𝑎⟩ ∈ ((Base‘𝑅) × (RLReg‘𝑅)) → [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) ∈ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
126	123, 125	syl 17	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) ∈ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
127	39	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (RLReg‘𝑅) ⊆ (Base‘𝑅))
128	16, 6, 17, 37, 62, 81, 61, 2, 127	rlocbas 33239	. . . . . . . . 9 ⊢ (𝜑 → (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅))))
129	128	ad4antr 731	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅))))
130	126, 129	eleqtrd 2846	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) ∈ (Base‘(𝑅 RLocal (RLReg‘𝑅))))
131		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘(𝑅 RLocal (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅)))
132		eqid 2740	. . . . . . . . . 10 ⊢ (.r‘(𝑅 RLocal (RLReg‘𝑅))) = (.r‘(𝑅 RLocal (RLReg‘𝑅)))
133		eqid 2740	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
134	16, 17, 133, 81, 61, 63, 71	rloccring 33242	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ CRing)
135	134	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑅 RLocal (RLReg‘𝑅)) ∈ CRing)
136		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}))
137	136	eldifad 3988	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑥 ∈ (Base‘(𝑅 RLocal (RLReg‘𝑅))))
138	131, 132, 135, 137, 130	crngcomd 20282	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))) = ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥))
139		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅)))
140	139	oveq2d 7464	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))))
141	63	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑅 ∈ CRing)
142	71	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (RLReg‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅)))
143	16, 17, 133, 81, 61, 141, 142, 93, 94, 122, 92, 132	rlocmulval 33241	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) = [⟨(𝑏(.r‘𝑅)𝑎), (𝑎(.r‘𝑅)𝑏)⟩](𝑅 ~RL (RLReg‘𝑅)))
144	72	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑅 ~RL (RLReg‘𝑅)) Er ((Base‘𝑅) × (RLReg‘𝑅)))
145	16, 17, 141, 93, 94	crngcomd 20282	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑏(.r‘𝑅)𝑎) = (𝑎(.r‘𝑅)𝑏))
146	18	ad4antr 731	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑅 ∈ Ring)
147	16, 17, 146, 93, 94	ringcld 20286	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑏(.r‘𝑅)𝑎) ∈ (Base‘𝑅))
148	16, 17, 5, 146, 147	ringridmd 20296	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((𝑏(.r‘𝑅)𝑎)(.r‘𝑅)(1r‘𝑅)) = (𝑏(.r‘𝑅)𝑎))
149	16, 17, 146, 94, 93	ringcld 20286	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝑅))
150	16, 17, 5, 146, 149	ringlidmd 20295	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((1r‘𝑅)(.r‘𝑅)(𝑎(.r‘𝑅)𝑏)) = (𝑎(.r‘𝑅)𝑏))
151	145, 148, 150	3eqtr4d 2790	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((𝑏(.r‘𝑅)𝑎)(.r‘𝑅)(1r‘𝑅)) = ((1r‘𝑅)(.r‘𝑅)(𝑎(.r‘𝑅)𝑏)))
152	73	ad4antr 731	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (1r‘𝑅) ∈ (Base‘𝑅))
153	94	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑎 ∈ (Base‘𝑅))
154	31	ad5antr 733	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → (0g‘𝑅) ∈ (Base‘𝑅))
155	92	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑏 ∈ (RLReg‘𝑅))
156	66	ad5antr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅))
157	155, 156	eleqtrrd 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
158	2	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑅 ∈ IDomn)
159	158	ad4antr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑅 ∈ IDomn)
160		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅))
161	146	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑅 ∈ Ring)
162	93	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑏 ∈ (Base‘𝑅))
163	16, 17, 6, 161, 162	ringlzd 20318	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑏) = (0g‘𝑅))
164	160, 163	eqtr4d 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → (𝑎(.r‘𝑅)𝑏) = ((0g‘𝑅)(.r‘𝑅)𝑏))
165	16, 6, 17, 153, 154, 157, 159, 164	idomrcan 33248	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑎 = (0g‘𝑅))
166	118, 165	mtand 815	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ¬ (𝑎(.r‘𝑅)𝑏) = (0g‘𝑅))
167	166	neqned 2953	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r‘𝑅)𝑏) ≠ (0g‘𝑅))
168	149, 167	eldifsnd 4812	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r‘𝑅)𝑏) ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
169	168, 121	eleqtrd 2846	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r‘𝑅)𝑏) ∈ (RLReg‘𝑅))
170	74	ad4antr 731	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (1r‘𝑅) ∈ (RLReg‘𝑅))
171	16, 17, 61, 141, 147, 152, 169, 170	fracerl 33273	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (⟨(𝑏(.r‘𝑅)𝑎), (𝑎(.r‘𝑅)𝑏)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(1r‘𝑅), (1r‘𝑅)⟩ ↔ ((𝑏(.r‘𝑅)𝑎)(.r‘𝑅)(1r‘𝑅)) = ((1r‘𝑅)(.r‘𝑅)(𝑎(.r‘𝑅)𝑏))))
172	151, 171	mpbird 257	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ⟨(𝑏(.r‘𝑅)𝑎), (𝑎(.r‘𝑅)𝑏)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(1r‘𝑅), (1r‘𝑅)⟩)
173	144, 172	erthi 8816	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → [⟨(𝑏(.r‘𝑅)𝑎), (𝑎(.r‘𝑅)𝑏)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
174	143, 173	eqtrd 2780	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) = [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
175	83	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
176	140, 174, 175	3eqtrd 2784	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
177	138, 176	eqtrd 2780	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
178	177, 176	jca 511	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))[⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ∧ ([⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅))(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))))
179	91, 130, 178	rspcedvdw 3638	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ∃𝑦 ∈ (Base‘(𝑅 RLocal (RLReg‘𝑅)))((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑦) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ∧ (𝑦(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))))
180	128	difeq1d 4148	. . . . . . . . . 10 ⊢ (𝜑 → ((((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}) = ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}))
181	180	eleq2d 2830	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ((((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}) ↔ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})))
182	181	biimpar 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑥 ∈ ((((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}))
183	182	eldifad 3988	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑥 ∈ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
184	183	elrlocbasi 33238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (RLReg‘𝑅)𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅)))
185	179, 184	r19.29vva 3222	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → ∃𝑦 ∈ (Base‘(𝑅 RLocal (RLReg‘𝑅)))((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑦) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ∧ (𝑦(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))))
186	185	ralrimiva 3152	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})∃𝑦 ∈ (Base‘(𝑅 RLocal (RLReg‘𝑅)))((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑦) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ∧ (𝑦(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))))
187		eqid 2740	. . . . 5 ⊢ (0g‘(𝑅 RLocal (RLReg‘𝑅))) = (0g‘(𝑅 RLocal (RLReg‘𝑅)))
188		eqid 2740	. . . . 5 ⊢ (1r‘(𝑅 RLocal (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))
189		eqid 2740	. . . . 5 ⊢ (Unit‘(𝑅 RLocal (RLReg‘𝑅))) = (Unit‘(𝑅 RLocal (RLReg‘𝑅)))
190	134	crngringd 20273	. . . . 5 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ Ring)
191	131, 187, 188, 132, 189, 190	isdrng4 33264	. . . 4 ⊢ (𝜑 → ((𝑅 RLocal (RLReg‘𝑅)) ∈ DivRing ↔ ((1r‘(𝑅 RLocal (RLReg‘𝑅))) ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))) ∧ ∀𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})∃𝑦 ∈ (Base‘(𝑅 RLocal (RLReg‘𝑅)))((𝑥(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑦) = (1r‘(𝑅 RLocal (RLReg‘𝑅))) ∧ (𝑦(.r‘(𝑅 RLocal (RLReg‘𝑅)))𝑥) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))))))
192	86, 186, 191	mpbir2and 712	. . 3 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ DivRing)
193		isfld 20762	. . 3 ⊢ ((𝑅 RLocal (RLReg‘𝑅)) ∈ Field ↔ ((𝑅 RLocal (RLReg‘𝑅)) ∈ DivRing ∧ (𝑅 RLocal (RLReg‘𝑅)) ∈ CRing))
194	192, 134, 193	sylanbrc 582	. 2 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ Field)
195	1, 194	eqeltrid 2848	1 ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ⊆ wss 3976  {csn 4648  ⟨cop 4654   class class class wbr 5166   × cxp 5698  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029   Er wer 8760  [cec 8761   / cqs 8762  Basecbs 17258  +_gcplusg 17311  ._rcmulr 17312  0_gc0g 17499  SubMndcsubmnd 18817  Grpcgrp 18973  -_gcsg 18975  mulGrpcmgp 20161  1_rcur 20208  Ringcrg 20260  CRingccrg 20261  Unitcui 20381  NzRingcnzr 20538  RLRegcrlreg 20713  Domncdomn 20714  IDomncidom 20715  DivRingcdr 20751  Fieldcfield 20752   ~_RL cerl 33225   RLocal crloc 33226   Frac cfrac 33269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-ec 8765  df-qs 8769  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-0g 17501  df-imas 17568  df-qus 17569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-nzr 20539  df-rlreg 20716  df-domn 20717  df-idom 20718  df-drng 20753  df-field 20754  df-erl 33227  df-rloc 33228  df-frac 33270

This theorem is referenced by:  idomsubr  33276  zringfrac  33547