Theorem fracfld 33495

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 33491	. 2 ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
2		fracfld.1	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ IDomn)
3	2	idomdomd 20776	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Domn)
4		domnnzr 20756	. . . . . . 7 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
5		eqid 2762	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
6		eqid 2762	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
7	5, 6	nzrnz 20565	. . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅))
8	3, 4, 7	3syl 18	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))
9		fvex 6880	. . . . . . . . . . . . . . . . 17 ⊢ (1r‘𝑅) ∈ V
10	9, 9	op1st 7978	. . . . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) ∈ V
13	12, 9	op2nd 7979	. . . . . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑅) = (.r‘𝑅)
18	2	idomringd 20778	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
19	18	ad2antrr 736	. . . . . . . . . . . . . . 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 20315	. . . . . . . . . . . . . . . 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 20322	. . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . 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 7978	. . . . . . . . . . . . . . . 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 7979	. . . . . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 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 20292	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ Grp)
30	16, 6	grpidcl 19007	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ (Base‘𝑅))
31	29, 30	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
32	31	ad2antrr 736	. . . . . . . . . . . . . . 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 20323	. . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . 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 7412	. . . . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
38		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
39	38, 16	rrgss 20752	. . . . . . . . . . . . . 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 3936	. . . . . . . . . . . 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 20357	. . . . . . . . . . 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 20323	. . . . . . . . . . . . 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 20346	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . 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 736	. . . . . . . . . . . . 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 19067	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑡 ∈ (Base‘𝑅)) → (𝑡(-g‘𝑅)(0g‘𝑅)) = 𝑡)
48	46, 41, 47	syl2anc 593	. . . . . . . . . . . 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 2797	. . . . . . . . . . 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 2801	. . . . . . . . . 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 2764	. . . . . . . . 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 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))
53		simpr 488	. . . . . . . . . 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 20754	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ NzRing → ¬ (0g‘𝑅) ∈ (RLReg‘𝑅))
55	3, 4, 54	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (0g‘𝑅) ∈ (RLReg‘𝑅))
56	55	ad2antrr 736	. . . . . . . . . 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 3055	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (RLReg‘𝑅) ∧ ¬ (0g‘𝑅) ∈ (RLReg‘𝑅)) → 𝑡 ≠ (0g‘𝑅))
58	53, 56, 57	syl2anc 593	. . . . . . . . 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 484	. . . . . . . 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 3041	. . . . . . 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 2762	. . . . . . . 8 ⊢ (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
62		eqid 2762	. . . . . . . . . . 11 ⊢ ((Base‘𝑅) × (RLReg‘𝑅)) = ((Base‘𝑅) × (RLReg‘𝑅))
63	2	idomcringd 20777	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ CRing)
64	16, 38, 6	isdomn6 20764	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅)))
65	3, 64	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ NzRing ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅)))
66	65	simprd 499	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (RLReg‘𝑅))
67		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
68	16, 6, 67	isdomn3 20765	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ Ring ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅))))
69	3, 68	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅))))
70	69	simprd 499	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅)))
71	66, 70	eqeltrrd 2863	. . . . . . . . . . 11 ⊢ (𝜑 → (RLReg‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅)))
72	16, 6, 5, 17, 37, 62, 61, 63, 71	erler 33446	. . . . . . . . . 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 33467	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝑅) ∈ (RLReg‘𝑅))
75	73, 74	opelxpd 5686	. . . . . . . . . 10 ⊢ (𝜑 → ⟨(1r‘𝑅), (1r‘𝑅)⟩ ∈ ((Base‘𝑅) × (RLReg‘𝑅)))
76	72, 75	erth 8733	. . . . . . . . 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 481	. . . . . . . 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 33443	. . . . . . 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 3170	. . . . . 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 3048	. . . . 5 ⊢ (𝜑 → [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) ≠ [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
81		eqid 2762	. . . . . 6 ⊢ (𝑅 RLocal (RLReg‘𝑅)) = (𝑅 RLocal (RLReg‘𝑅))
82		eqid 2762	. . . . . 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 33454	. . . . 5 ⊢ (𝜑 → [⟨(1r‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
84		eqid 2762	. . . . . 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 33453	. . . . 5 ⊢ (𝜑 → [⟨(0g‘𝑅), (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
86	80, 83, 85	3netr3d 3033	. . . 4 ⊢ (𝜑 → (1r‘(𝑅 RLocal (RLReg‘𝑅))) ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
87		oveq2 7404	. . . . . . . . 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 2764	. . . . . . . 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 7403	. . . . . . . . 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 2764	. . . . . . . 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 641	. . . . . . 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 778	. . . . . . . . . . 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 3934	. . . . . . . . . 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 785	. . . . . . . . . . . 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 778	. . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . . . . 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 20345	. . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . 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 7411	. . . . . . . . . . . . . . . . . 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 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)) → 𝑏 ∈ (Base‘𝑅))
103	16, 17, 6, 97, 102	ringlzd 20345	. . . . . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . . . 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 33493	. . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . 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 8735	. . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . 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 4750	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}) → 𝑥 ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
116	115	ad5antlr 745	. . . . . . . . . . . . . . 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 2962	. . . . . . . . . . . . . 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 826	. . . . . . . . . . . . 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 2964	. . . . . . . . . . . 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 4747	. . . . . . . . . . 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 742	. . . . . . . . . . 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 2864	. . . . . . . . . 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 5686	. . . . . . . . 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 7429	. . . . . . . . . 10 ⊢ (𝑅 ~RL (RLReg‘𝑅)) ∈ V
125	124	ecelqsi 8751	. . . . . . . . 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 33449	. . . . . . . . 9 ⊢ (𝜑 → (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅))))
129	128	ad4antr 742	. . . . . . . 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 2864	. . . . . . 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 2762	. . . . . . . . . 10 ⊢ (Base‘(𝑅 RLocal (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅)))
132		eqid 2762	. . . . . . . . . 10 ⊢ (.r‘(𝑅 RLocal (RLReg‘𝑅))) = (.r‘(𝑅 RLocal (RLReg‘𝑅)))
133		eqid 2762	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
134	16, 17, 133, 81, 61, 63, 71	rloccring 33452	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ CRing)
135	134	ad4antr 742	. . . . . . . . . 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 793	. . . . . . . . . . 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 3916	. . . . . . . . . 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 20305	. . . . . . . . 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 488	. . . . . . . . . . 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 7412	. . . . . . . . . 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 742	. . . . . . . . . . . 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 742	. . . . . . . . . . . 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 33451	. . . . . . . . . . 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 742	. . . . . . . . . . . 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 20305	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . 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 20310	. . . . . . . . . . . . . . 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 20323	. . . . . . . . . . . . . 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 20310	. . . . . . . . . . . . . . 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 20322	. . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . 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 742	. . . . . . . . . . . . . 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 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(.r‘𝑅)𝑏) = (0g‘𝑅)) → 𝑎 ∈ (Base‘𝑅))
154	31	ad5antr 744	. . . . . . . . . . . . . . . . . . 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 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)) → 𝑏 ∈ (RLReg‘𝑅))
156	66	ad5antr 744	. . . . . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑅 ∈ IDomn)
159	158	ad4antr 742	. . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . 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 20345	. . . . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . . . 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 33463	. . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . 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 2964	. . . . . . . . . . . . . . . 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 4747	. . . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . 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 33493	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 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 8735	. . . . . . . . . . 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 2797	. . . . . . . . . 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 742	. . . . . . . . . 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 2801	. . . . . . . . 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 2797	. . . . . . . 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 519	. . . . . . 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 3584	. . . . . 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 4079	. . . . . . . . . 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 2848	. . . . . . . . 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 481	. . . . . . . 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 3916	. . . . . . 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 33448	. . . . . 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 3154	. . . 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 2762	. . . . 5 ⊢ (0g‘(𝑅 RLocal (RLReg‘𝑅))) = (0g‘(𝑅 RLocal (RLReg‘𝑅)))
188		eqid 2762	. . . . 5 ⊢ (1r‘(𝑅 RLocal (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))
189		eqid 2762	. . . . 5 ⊢ (Unit‘(𝑅 RLocal (RLReg‘𝑅))) = (Unit‘(𝑅 RLocal (RLReg‘𝑅)))
190	134	crngringd 20296	. . . . 5 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ Ring)
191	131, 187, 188, 132, 189, 190	isdrng4 33482	. . . 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 723	. . 3 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ DivRing)
193		isfld 20790	. . 3 ⊢ ((𝑅 RLocal (RLReg‘𝑅)) ∈ Field ↔ ((𝑅 RLocal (RLReg‘𝑅)) ∈ DivRing ∧ (𝑅 RLocal (RLReg‘𝑅)) ∈ CRing))
194	192, 134, 193	sylanbrc 592	. 2 ⊢ (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ Field)
195	1, 194	eqeltrid 2866	1 ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086   ∖ cdif 3901   ⊆ wss 3904  {csn 4582  ⟨cop 4588   class class class wbr 5100   × cxp 5645  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  2^nd c2nd 7969   Er wer 8675  [cec 8676   / cqs 8677  Basecbs 17245  +_gcplusg 17286  ._rcmulr 17287  0_gc0g 17468  SubMndcsubmnd 18816  Grpcgrp 18975  -_gcsg 18977  mulGrpcmgp 20186  1_rcur 20231  Ringcrg 20283  CRingccrg 20284  Unitcui 20404  NzRingcnzr 20562  RLRegcrlreg 20741  Domncdomn 20742  IDomncidom 20743  DivRingcdr 20779  Fieldcfield 20780   ~_RL cerl 33434   RLocal crloc 33435   Frac cfrac 33489

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

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-ec 8680  df-qs 8684  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-inf 9389  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-0g 17470  df-imas 17538  df-qus 17539  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-submnd 18818  df-grp 18978  df-minusg 18979  df-sbg 18980  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20232  df-ring 20285  df-cring 20286  df-oppr 20386  df-dvdsr 20406  df-unit 20407  df-invr 20437  df-nzr 20563  df-rlreg 20744  df-domn 20745  df-idom 20746  df-drng 20781  df-field 20782  df-erl 33436  df-rloc 33437  df-frac 33490

This theorem is referenced by:  idomsubr  33496  zringfrac  33750