Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fracfld Structured version   Visualization version   GIF version

Theorem fracfld 33311
Description: The field of fractions of an integral domain is a field. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypothesis
Ref Expression
fracfld.1 (𝜑𝑅 ∈ 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.
StepHypRef Expression
1 fracval 33307 . 2 ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
2 fracfld.1 . . . . . . . 8 (𝜑𝑅 ∈ IDomn)
32idomdomd 20727 . . . . . . 7 (𝜑𝑅 ∈ Domn)
4 domnnzr 20707 . . . . . . 7 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
5 eqid 2736 . . . . . . . 8 (1r𝑅) = (1r𝑅)
6 eqid 2736 . . . . . . . 8 (0g𝑅) = (0g𝑅)
75, 6nzrnz 20516 . . . . . . 7 (𝑅 ∈ NzRing → (1r𝑅) ≠ (0g𝑅))
83, 4, 73syl 18 . . . . . 6 (𝜑 → (1r𝑅) ≠ (0g𝑅))
9 fvex 6918 . . . . . . . . . . . . . . . . 17 (1r𝑅) ∈ V
109, 9op1st 8023 . . . . . . . . . . . . . . . 16 (1st ‘⟨(1r𝑅), (1r𝑅)⟩) = (1r𝑅)
1110a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (1st ‘⟨(1r𝑅), (1r𝑅)⟩) = (1r𝑅))
12 fvex 6918 . . . . . . . . . . . . . . . . 17 (0g𝑅) ∈ V
1312, 9op2nd 8024 . . . . . . . . . . . . . . . 16 (2nd ‘⟨(0g𝑅), (1r𝑅)⟩) = (1r𝑅)
1413a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (2nd ‘⟨(0g𝑅), (1r𝑅)⟩) = (1r𝑅))
1511, 14oveq12d 7450 . . . . . . . . . . . . . 14 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩)) = ((1r𝑅)(.r𝑅)(1r𝑅)))
16 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
17 eqid 2736 . . . . . . . . . . . . . . 15 (.r𝑅) = (.r𝑅)
182idomringd 20729 . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ Ring)
1918ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑅 ∈ Ring)
2016, 5ringidcl 20263 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
2119, 20syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (1r𝑅) ∈ (Base‘𝑅))
2216, 17, 5, 19, 21ringlidmd 20270 . . . . . . . . . . . . . 14 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1r𝑅)(.r𝑅)(1r𝑅)) = (1r𝑅))
2315, 22eqtrd 2776 . . . . . . . . . . . . 13 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩)) = (1r𝑅))
2412, 9op1st 8023 . . . . . . . . . . . . . . . 16 (1st ‘⟨(0g𝑅), (1r𝑅)⟩) = (0g𝑅)
2524a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (1st ‘⟨(0g𝑅), (1r𝑅)⟩) = (0g𝑅))
269, 9op2nd 8024 . . . . . . . . . . . . . . . 16 (2nd ‘⟨(1r𝑅), (1r𝑅)⟩) = (1r𝑅)
2726a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (2nd ‘⟨(1r𝑅), (1r𝑅)⟩) = (1r𝑅))
2825, 27oveq12d 7450 . . . . . . . . . . . . . 14 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)) = ((0g𝑅)(.r𝑅)(1r𝑅)))
2918ringgrpd 20240 . . . . . . . . . . . . . . . . 17 (𝜑𝑅 ∈ Grp)
3016, 6grpidcl 18984 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Grp → (0g𝑅) ∈ (Base‘𝑅))
3129, 30syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝑅) ∈ (Base‘𝑅))
3231ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (0g𝑅) ∈ (Base‘𝑅))
3316, 17, 5, 19, 32ringridmd 20271 . . . . . . . . . . . . . 14 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((0g𝑅)(.r𝑅)(1r𝑅)) = (0g𝑅))
3428, 33eqtrd 2776 . . . . . . . . . . . . 13 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)) = (0g𝑅))
3523, 34oveq12d 7450 . . . . . . . . . . . 12 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩))) = ((1r𝑅)(-g𝑅)(0g𝑅)))
3635oveq2d 7448 . . . . . . . . . . 11 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r𝑅)(((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)))) = (𝑡(.r𝑅)((1r𝑅)(-g𝑅)(0g𝑅))))
37 eqid 2736 . . . . . . . . . . . 12 (-g𝑅) = (-g𝑅)
38 eqid 2736 . . . . . . . . . . . . . . 15 (RLReg‘𝑅) = (RLReg‘𝑅)
3938, 16rrgss 20703 . . . . . . . . . . . . . 14 (RLReg‘𝑅) ⊆ (Base‘𝑅)
4039a1i 11 . . . . . . . . . . . . 13 ((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (RLReg‘𝑅) ⊆ (Base‘𝑅))
4140sselda 3982 . . . . . . . . . . . 12 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑡 ∈ (Base‘𝑅))
4216, 17, 37, 19, 41, 21, 32ringsubdi 20305 . . . . . . . . . . 11 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r𝑅)((1r𝑅)(-g𝑅)(0g𝑅))) = ((𝑡(.r𝑅)(1r𝑅))(-g𝑅)(𝑡(.r𝑅)(0g𝑅))))
4316, 17, 5, 19, 41ringridmd 20271 . . . . . . . . . . . . 13 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r𝑅)(1r𝑅)) = 𝑡)
4416, 17, 6, 19, 41ringrzd 20294 . . . . . . . . . . . . 13 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r𝑅)(0g𝑅)) = (0g𝑅))
4543, 44oveq12d 7450 . . . . . . . . . . . 12 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((𝑡(.r𝑅)(1r𝑅))(-g𝑅)(𝑡(.r𝑅)(0g𝑅))) = (𝑡(-g𝑅)(0g𝑅)))
4629ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑅 ∈ Grp)
4716, 6, 37grpsubid1 19044 . . . . . . . . . . . . 13 ((𝑅 ∈ Grp ∧ 𝑡 ∈ (Base‘𝑅)) → (𝑡(-g𝑅)(0g𝑅)) = 𝑡)
4846, 41, 47syl2anc 584 . . . . . . . . . . . 12 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(-g𝑅)(0g𝑅)) = 𝑡)
4945, 48eqtrd 2776 . . . . . . . . . . 11 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((𝑡(.r𝑅)(1r𝑅))(-g𝑅)(𝑡(.r𝑅)(0g𝑅))) = 𝑡)
5036, 42, 493eqtrd 2780 . . . . . . . . . 10 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r𝑅)(((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)))) = 𝑡)
5150eqeq1d 2738 . . . . . . . . 9 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((𝑡(.r𝑅)(((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)))) = (0g𝑅) ↔ 𝑡 = (0g𝑅)))
5251biimpa 476 . . . . . . . 8 ((((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡(.r𝑅)(((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)))) = (0g𝑅)) → 𝑡 = (0g𝑅))
53 simpr 484 . . . . . . . . . 10 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑡 ∈ (RLReg‘𝑅))
5438, 6rrgnz 20705 . . . . . . . . . . . 12 (𝑅 ∈ NzRing → ¬ (0g𝑅) ∈ (RLReg‘𝑅))
553, 4, 543syl 18 . . . . . . . . . . 11 (𝜑 → ¬ (0g𝑅) ∈ (RLReg‘𝑅))
5655ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ¬ (0g𝑅) ∈ (RLReg‘𝑅))
57 nelne2 3039 . . . . . . . . . 10 ((𝑡 ∈ (RLReg‘𝑅) ∧ ¬ (0g𝑅) ∈ (RLReg‘𝑅)) → 𝑡 ≠ (0g𝑅))
5853, 56, 57syl2anc 584 . . . . . . . . 9 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → 𝑡 ≠ (0g𝑅))
5958adantr 480 . . . . . . . 8 ((((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡(.r𝑅)(((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)))) = (0g𝑅)) → 𝑡 ≠ (0g𝑅))
6052, 59pm2.21ddne 3025 . . . . . . 7 ((((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡(.r𝑅)(((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)))) = (0g𝑅)) → (1r𝑅) = (0g𝑅))
61 eqid 2736 . . . . . . . 8 (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
62 eqid 2736 . . . . . . . . . . 11 ((Base‘𝑅) × (RLReg‘𝑅)) = ((Base‘𝑅) × (RLReg‘𝑅))
632idomcringd 20728 . . . . . . . . . . 11 (𝜑𝑅 ∈ CRing)
6416, 38, 6isdomn6 20715 . . . . . . . . . . . . . 14 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ((Base‘𝑅) ∖ {(0g𝑅)}) = (RLReg‘𝑅)))
653, 64sylib 218 . . . . . . . . . . . . 13 (𝜑 → (𝑅 ∈ NzRing ∧ ((Base‘𝑅) ∖ {(0g𝑅)}) = (RLReg‘𝑅)))
6665simprd 495 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝑅) ∖ {(0g𝑅)}) = (RLReg‘𝑅))
67 eqid 2736 . . . . . . . . . . . . . . 15 (mulGrp‘𝑅) = (mulGrp‘𝑅)
6816, 6, 67isdomn3 20716 . . . . . . . . . . . . . 14 (𝑅 ∈ Domn ↔ (𝑅 ∈ Ring ∧ ((Base‘𝑅) ∖ {(0g𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅))))
693, 68sylib 218 . . . . . . . . . . . . 13 (𝜑 → (𝑅 ∈ Ring ∧ ((Base‘𝑅) ∖ {(0g𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅))))
7069simprd 495 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝑅) ∖ {(0g𝑅)}) ∈ (SubMnd‘(mulGrp‘𝑅)))
7166, 70eqeltrrd 2841 . . . . . . . . . . 11 (𝜑 → (RLReg‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅)))
7216, 6, 5, 17, 37, 62, 61, 63, 71erler 33270 . . . . . . . . . 10 (𝜑 → (𝑅 ~RL (RLReg‘𝑅)) Er ((Base‘𝑅) × (RLReg‘𝑅)))
7318, 20syl 17 . . . . . . . . . . 11 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
745, 38, 181rrg 33287 . . . . . . . . . . 11 (𝜑 → (1r𝑅) ∈ (RLReg‘𝑅))
7573, 74opelxpd 5723 . . . . . . . . . 10 (𝜑 → ⟨(1r𝑅), (1r𝑅)⟩ ∈ ((Base‘𝑅) × (RLReg‘𝑅)))
7672, 75erth 8797 . . . . . . . . 9 (𝜑 → (⟨(1r𝑅), (1r𝑅)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g𝑅), (1r𝑅)⟩ ↔ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
7776biimpar 477 . . . . . . . 8 ((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → ⟨(1r𝑅), (1r𝑅)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g𝑅), (1r𝑅)⟩)
7816, 61, 40, 6, 17, 37, 77erldi 33267 . . . . . . 7 ((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → ∃𝑡 ∈ (RLReg‘𝑅)(𝑡(.r𝑅)(((1st ‘⟨(1r𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(0g𝑅), (1r𝑅)⟩))(-g𝑅)((1st ‘⟨(0g𝑅), (1r𝑅)⟩)(.r𝑅)(2nd ‘⟨(1r𝑅), (1r𝑅)⟩)))) = (0g𝑅))
7960, 78r19.29a 3161 . . . . . 6 ((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (1r𝑅) = (0g𝑅))
808, 79mteqand 3032 . . . . 5 (𝜑 → [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) ≠ [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
81 eqid 2736 . . . . . 6 (𝑅 RLocal (RLReg‘𝑅)) = (𝑅 RLocal (RLReg‘𝑅))
82 eqid 2736 . . . . . 6 [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))
836, 5, 81, 61, 63, 71, 82rloc1r 33277 . . . . 5 (𝜑 → [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
84 eqid 2736 . . . . . 6 [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))
856, 5, 81, 61, 63, 71, 84rloc0g 33276 . . . . 5 (𝜑 → [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
8680, 83, 853netr3d 3016 . . . 4 (𝜑 → (1r‘(𝑅 RLocal (RLReg‘𝑅))) ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
87 oveq2 7440 . . . . . . . . 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))))
8887eqeq1d 2738 . . . . . . . 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 7439 . . . . . . . . 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))
9089eqeq1d 2738 . . . . . . . 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)))))
9188, 90anbi12d 632 . . . . . . 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 (((((𝜑𝑥 ∈ ((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))
9339, 92sselid 3980 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ((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 (((((𝜑𝑥 ∈ ((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 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅)))
9672ad5antr 734 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → (𝑅 ~RL (RLReg‘𝑅)) Er ((Base‘𝑅) × (RLReg‘𝑅)))
9718ad5antr 734 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑅 ∈ Ring)
9897, 20syl 17 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → (1r𝑅) ∈ (Base‘𝑅))
9916, 17, 6, 97, 98ringlzd 20293 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → ((0g𝑅)(.r𝑅)(1r𝑅)) = (0g𝑅))
100 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑎 = (0g𝑅))
101100oveq1d 7447 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → (𝑎(.r𝑅)(1r𝑅)) = ((0g𝑅)(.r𝑅)(1r𝑅)))
10293adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑏 ∈ (Base‘𝑅))
10316, 17, 6, 97, 102ringlzd 20293 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → ((0g𝑅)(.r𝑅)𝑏) = (0g𝑅))
10499, 101, 1033eqtr4d 2786 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → (𝑎(.r𝑅)(1r𝑅)) = ((0g𝑅)(.r𝑅)𝑏))
10563ad5antr 734 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑅 ∈ CRing)
10694adantr 480 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑎 ∈ (Base‘𝑅))
10731ad5antr 734 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → (0g𝑅) ∈ (Base‘𝑅))
10892adantr 480 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑏 ∈ (RLReg‘𝑅))
10974ad5antr 734 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → (1r𝑅) ∈ (RLReg‘𝑅))
11016, 17, 61, 105, 106, 107, 108, 109fracerl 33309 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → (⟨𝑎, 𝑏⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g𝑅), (1r𝑅)⟩ ↔ (𝑎(.r𝑅)(1r𝑅)) = ((0g𝑅)(.r𝑅)𝑏)))
111104, 110mpbird 257 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → ⟨𝑎, 𝑏⟩(𝑅 ~RL (RLReg‘𝑅))⟨(0g𝑅), (1r𝑅)⟩)
11296, 111erthi 8799 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
11385ad5antr 734 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
11495, 112, 1133eqtrd 2780 . . . . . . . . . . . . . 14 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑥 = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
115 eldifsni 4789 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}) → 𝑥 ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
116115ad5antlr 735 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → 𝑥 ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
117116neneqd 2944 . . . . . . . . . . . . . 14 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑎 = (0g𝑅)) → ¬ 𝑥 = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
118114, 117pm2.65da 816 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ¬ 𝑎 = (0g𝑅))
119118neqned 2946 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → 𝑎 ≠ (0g𝑅))
12094, 119eldifsnd 4786 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ((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) ∖ {(0g𝑅)}))
12166ad4antr 732 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ((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) ∖ {(0g𝑅)}) = (RLReg‘𝑅))
122120, 121eleqtrd 2842 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ((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))
12393, 122opelxpd 5723 . . . . . . . . 9 (((((𝜑𝑥 ∈ ((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 7465 . . . . . . . . . 10 (𝑅 ~RL (RLReg‘𝑅)) ∈ V
125124ecelqsi 8814 . . . . . . . . 9 (⟨𝑏, 𝑎⟩ ∈ ((Base‘𝑅) × (RLReg‘𝑅)) → [⟨𝑏, 𝑎⟩](𝑅 ~RL (RLReg‘𝑅)) ∈ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
126123, 125syl 17 . . . . . . . 8 (((((𝜑𝑥 ∈ ((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))))
12739a1i 11 . . . . . . . . . 10 (𝜑 → (RLReg‘𝑅) ⊆ (Base‘𝑅))
12816, 6, 17, 37, 62, 81, 61, 2, 127rlocbas 33272 . . . . . . . . 9 (𝜑 → (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅))))
129128ad4antr 732 . . . . . . . 8 (((((𝜑𝑥 ∈ ((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))))
130126, 129eleqtrd 2842 . . . . . . 7 (((((𝜑𝑥 ∈ ((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 2736 . . . . . . . . . 10 (Base‘(𝑅 RLocal (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅)))
132 eqid 2736 . . . . . . . . . 10 (.r‘(𝑅 RLocal (RLReg‘𝑅))) = (.r‘(𝑅 RLocal (RLReg‘𝑅)))
133 eqid 2736 . . . . . . . . . . . 12 (+g𝑅) = (+g𝑅)
13416, 17, 133, 81, 61, 63, 71rloccring 33275 . . . . . . . . . . 11 (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ CRing)
135134ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ((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 (((((𝜑𝑥 ∈ ((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)))}))
137136eldifad 3962 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ((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))))
138131, 132, 135, 137, 130crngcomd 20253 . . . . . . . . 9 (((((𝜑𝑥 ∈ ((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 (((((𝜑𝑥 ∈ ((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)))
140139oveq2d 7448 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ((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))))
14163ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ((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)
14271ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ((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)))
14316, 17, 133, 81, 61, 141, 142, 93, 94, 122, 92, 132rlocmulval 33274 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ((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(.r𝑅)𝑎), (𝑎(.r𝑅)𝑏)⟩](𝑅 ~RL (RLReg‘𝑅)))
14472ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ((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)))
14516, 17, 141, 93, 94crngcomd 20253 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑏(.r𝑅)𝑎) = (𝑎(.r𝑅)𝑏))
14618ad4antr 732 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ((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)
14716, 17, 146, 93, 94ringcld 20258 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑏(.r𝑅)𝑎) ∈ (Base‘𝑅))
14816, 17, 5, 146, 147ringridmd 20271 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((𝑏(.r𝑅)𝑎)(.r𝑅)(1r𝑅)) = (𝑏(.r𝑅)𝑎))
14916, 17, 146, 94, 93ringcld 20258 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r𝑅)𝑏) ∈ (Base‘𝑅))
15016, 17, 5, 146, 149ringlidmd 20270 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((1r𝑅)(.r𝑅)(𝑎(.r𝑅)𝑏)) = (𝑎(.r𝑅)𝑏))
151145, 148, 1503eqtr4d 2786 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ((𝑏(.r𝑅)𝑎)(.r𝑅)(1r𝑅)) = ((1r𝑅)(.r𝑅)(𝑎(.r𝑅)𝑏)))
15273ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (1r𝑅) ∈ (Base‘𝑅))
15394adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → 𝑎 ∈ (Base‘𝑅))
15431ad5antr 734 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → (0g𝑅) ∈ (Base‘𝑅))
15592adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → 𝑏 ∈ (RLReg‘𝑅))
15666ad5antr 734 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → ((Base‘𝑅) ∖ {(0g𝑅)}) = (RLReg‘𝑅))
157155, 156eleqtrrd 2843 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}))
1582adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑅 ∈ IDomn)
159158ad4antr 732 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → 𝑅 ∈ IDomn)
160 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → (𝑎(.r𝑅)𝑏) = (0g𝑅))
161146adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → 𝑅 ∈ Ring)
16293adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → 𝑏 ∈ (Base‘𝑅))
16316, 17, 6, 161, 162ringlzd 20293 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → ((0g𝑅)(.r𝑅)𝑏) = (0g𝑅))
164160, 163eqtr4d 2779 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → (𝑎(.r𝑅)𝑏) = ((0g𝑅)(.r𝑅)𝑏))
16516, 6, 17, 153, 154, 157, 159, 164idomrcan 33283 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑎(.r𝑅)𝑏) = (0g𝑅)) → 𝑎 = (0g𝑅))
166118, 165mtand 815 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ¬ (𝑎(.r𝑅)𝑏) = (0g𝑅))
167166neqned 2946 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r𝑅)𝑏) ≠ (0g𝑅))
168149, 167eldifsnd 4786 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r𝑅)𝑏) ∈ ((Base‘𝑅) ∖ {(0g𝑅)}))
169168, 121eleqtrd 2842 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑎(.r𝑅)𝑏) ∈ (RLReg‘𝑅))
17074ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (1r𝑅) ∈ (RLReg‘𝑅))
17116, 17, 61, 141, 147, 152, 169, 170fracerl 33309 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → (⟨(𝑏(.r𝑅)𝑎), (𝑎(.r𝑅)𝑏)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(1r𝑅), (1r𝑅)⟩ ↔ ((𝑏(.r𝑅)𝑎)(.r𝑅)(1r𝑅)) = ((1r𝑅)(.r𝑅)(𝑎(.r𝑅)𝑏))))
172151, 171mpbird 257 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → ⟨(𝑏(.r𝑅)𝑎), (𝑎(.r𝑅)𝑏)⟩(𝑅 ~RL (RLReg‘𝑅))⟨(1r𝑅), (1r𝑅)⟩)
173144, 172erthi 8799 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → [⟨(𝑏(.r𝑅)𝑎), (𝑎(.r𝑅)𝑏)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
174143, 173eqtrd 2776 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ((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))) = [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
17583ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (RLReg‘𝑅)) ∧ 𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅))) → [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
176140, 174, 1753eqtrd 2780 . . . . . . . . 9 (((((𝜑𝑥 ∈ ((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))))
177138, 176eqtrd 2776 . . . . . . . 8 (((((𝜑𝑥 ∈ ((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))))
178177, 176jca 511 . . . . . . 7 (((((𝜑𝑥 ∈ ((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)))))
17991, 130, 178rspcedvdw 3624 . . . . . 6 (((((𝜑𝑥 ∈ ((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)))))
180128difeq1d 4124 . . . . . . . . . 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)))}))
181180eleq2d 2826 . . . . . . . . 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)))})))
182181biimpar 477 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑥 ∈ ((((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))}))
183182eldifad 3962 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑥 ∈ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
184183elrlocbasi 33271 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (RLReg‘𝑅)𝑥 = [⟨𝑎, 𝑏⟩](𝑅 ~RL (RLReg‘𝑅)))
185179, 184r19.29vva 3215 . . . . 5 ((𝜑𝑥 ∈ ((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)))))
186185ralrimiva 3145 . . . 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 2736 . . . . 5 (0g‘(𝑅 RLocal (RLReg‘𝑅))) = (0g‘(𝑅 RLocal (RLReg‘𝑅)))
188 eqid 2736 . . . . 5 (1r‘(𝑅 RLocal (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))
189 eqid 2736 . . . . 5 (Unit‘(𝑅 RLocal (RLReg‘𝑅))) = (Unit‘(𝑅 RLocal (RLReg‘𝑅)))
190134crngringd 20244 . . . . 5 (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ Ring)
191131, 187, 188, 132, 189, 190isdrng4 33299 . . . 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)))))))
19286, 186, 191mpbir2and 713 . . 3 (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ DivRing)
193 isfld 20741 . . 3 ((𝑅 RLocal (RLReg‘𝑅)) ∈ Field ↔ ((𝑅 RLocal (RLReg‘𝑅)) ∈ DivRing ∧ (𝑅 RLocal (RLReg‘𝑅)) ∈ CRing))
194192, 134, 193sylanbrc 583 . 2 (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ Field)
1951, 194eqeltrid 2844 1 (𝜑 → ( Frac ‘𝑅) ∈ Field)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  cdif 3947  wss 3950  {csn 4625  cop 4631   class class class wbr 5142   × cxp 5682  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014   Er wer 8743  [cec 8744   / cqs 8745  Basecbs 17248  +gcplusg 17298  .rcmulr 17299  0gc0g 17485  SubMndcsubmnd 18796  Grpcgrp 18952  -gcsg 18954  mulGrpcmgp 20138  1rcur 20179  Ringcrg 20231  CRingccrg 20232  Unitcui 20356  NzRingcnzr 20513  RLRegcrlreg 20692  Domncdomn 20693  IDomncidom 20694  DivRingcdr 20730  Fieldcfield 20731   ~RL cerl 33258   RLocal crloc 33259   Frac cfrac 33305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-ec 8748  df-qs 8752  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17487  df-imas 17554  df-qus 17555  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-nzr 20514  df-rlreg 20695  df-domn 20696  df-idom 20697  df-drng 20732  df-field 20733  df-erl 33260  df-rloc 33261  df-frac 33306
This theorem is referenced by:  idomsubr  33312  zringfrac  33583
  Copyright terms: Public domain W3C validator