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Theorem fracfld 33289
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 33285 . 2 ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
2 fracfld.1 . . . . . . . 8 (𝜑𝑅 ∈ IDomn)
32idomdomd 20742 . . . . . . 7 (𝜑𝑅 ∈ Domn)
4 domnnzr 20722 . . . . . . 7 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
5 eqid 2734 . . . . . . . 8 (1r𝑅) = (1r𝑅)
6 eqid 2734 . . . . . . . 8 (0g𝑅) = (0g𝑅)
75, 6nzrnz 20531 . . . . . . 7 (𝑅 ∈ NzRing → (1r𝑅) ≠ (0g𝑅))
83, 4, 73syl 18 . . . . . 6 (𝜑 → (1r𝑅) ≠ (0g𝑅))
9 fvex 6919 . . . . . . . . . . . . . . . . 17 (1r𝑅) ∈ V
109, 9op1st 8020 . . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . . 17 (0g𝑅) ∈ V
1312, 9op2nd 8021 . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
17 eqid 2734 . . . . . . . . . . . . . . 15 (.r𝑅) = (.r𝑅)
182idomringd 20744 . . . . . . . . . . . . . . . 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 20279 . . . . . . . . . . . . . . . 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 20285 . . . . . . . . . . . . . 14 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((1r𝑅)(.r𝑅)(1r𝑅)) = (1r𝑅))
2315, 22eqtrd 2774 . . . . . . . . . . . . 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 8020 . . . . . . . . . . . . . . . 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 8021 . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . 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 20259 . . . . . . . . . . . . . . . . 17 (𝜑𝑅 ∈ Grp)
3016, 6grpidcl 18995 . . . . . . . . . . . . . . . . 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 20286 . . . . . . . . . . . . . 14 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → ((0g𝑅)(.r𝑅)(1r𝑅)) = (0g𝑅))
3428, 33eqtrd 2774 . . . . . . . . . . . . 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 7448 . . . . . . . . . . . 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 7446 . . . . . . . . . . 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 2734 . . . . . . . . . . . 12 (-g𝑅) = (-g𝑅)
38 eqid 2734 . . . . . . . . . . . . . . 15 (RLReg‘𝑅) = (RLReg‘𝑅)
3938, 16rrgss 20718 . . . . . . . . . . . . . 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 3994 . . . . . . . . . . . 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 20320 . . . . . . . . . . 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 20286 . . . . . . . . . . . . 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 20309 . . . . . . . . . . . . 13 (((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∧ 𝑡 ∈ (RLReg‘𝑅)) → (𝑡(.r𝑅)(0g𝑅)) = (0g𝑅))
4543, 44oveq12d 7448 . . . . . . . . . . . 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 19055 . . . . . . . . . . . . 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 2774 . . . . . . . . . . 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 2778 . . . . . . . . . 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 2736 . . . . . . . . 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 20720 . . . . . . . . . . . 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 3037 . . . . . . . . . 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 3023 . . . . . . 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 2734 . . . . . . . 8 (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
62 eqid 2734 . . . . . . . . . . 11 ((Base‘𝑅) × (RLReg‘𝑅)) = ((Base‘𝑅) × (RLReg‘𝑅))
632idomcringd 20743 . . . . . . . . . . 11 (𝜑𝑅 ∈ CRing)
6416, 38, 6isdomn6 20730 . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . 15 (mulGrp‘𝑅) = (mulGrp‘𝑅)
6816, 6, 67isdomn3 20731 . . . . . . . . . . . . . 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 2839 . . . . . . . . . . 11 (𝜑 → (RLReg‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅)))
7216, 6, 5, 17, 37, 62, 61, 63, 71erler 33251 . . . . . . . . . 10 (𝜑 → (𝑅 ~RL (RLReg‘𝑅)) Er ((Base‘𝑅) × (RLReg‘𝑅)))
7318, 20syl 17 . . . . . . . . . . 11 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
745, 38, 181rrg 33266 . . . . . . . . . . 11 (𝜑 → (1r𝑅) ∈ (RLReg‘𝑅))
7573, 74opelxpd 5727 . . . . . . . . . 10 (𝜑 → ⟨(1r𝑅), (1r𝑅)⟩ ∈ ((Base‘𝑅) × (RLReg‘𝑅)))
7672, 75erth 8794 . . . . . . . . 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 33248 . . . . . . 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 3159 . . . . . 6 ((𝜑 ∧ [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (1r𝑅) = (0g𝑅))
808, 79mteqand 3030 . . . . 5 (𝜑 → [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) ≠ [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
81 eqid 2734 . . . . . 6 (𝑅 RLocal (RLReg‘𝑅)) = (𝑅 RLocal (RLReg‘𝑅))
82 eqid 2734 . . . . . 6 [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))
836, 5, 81, 61, 63, 71, 82rloc1r 33258 . . . . 5 (𝜑 → [⟨(1r𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (1r‘(𝑅 RLocal (RLReg‘𝑅))))
84 eqid 2734 . . . . . 6 [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))
856, 5, 81, 61, 63, 71, 84rloc0g 33257 . . . . 5 (𝜑 → [⟨(0g𝑅), (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = (0g‘(𝑅 RLocal (RLReg‘𝑅))))
8680, 83, 853netr3d 3014 . . . 4 (𝜑 → (1r‘(𝑅 RLocal (RLReg‘𝑅))) ≠ (0g‘(𝑅 RLocal (RLReg‘𝑅))))
87 oveq2 7438 . . . . . . . . 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 2736 . . . . . . . 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 7437 . . . . . . . . 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 2736 . . . . . . . 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 769 . . . . . . . . . . 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 3992 . . . . . . . . . 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 776 . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 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 20308 . . . . . . . . . . . . . . . . . 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 7445 . . . . . . . . . . . . . . . . . 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 20308 . . . . . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . . . . . . 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 33287 . . . . . . . . . . . . . . . . 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 8796 . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . 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 4794 . . . . . . . . . . . . . . . 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 2942 . . . . . . . . . . . . . 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 817 . . . . . . . . . . . . 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 2944 . . . . . . . . . . . 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 4791 . . . . . . . . . . 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 2840 . . . . . . . . . 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 5727 . . . . . . . . 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 7463 . . . . . . . . . 10 (𝑅 ~RL (RLReg‘𝑅)) ∈ V
125124ecelqsi 8811 . . . . . . . . 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 33253 . . . . . . . . 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 2840 . . . . . . 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 2734 . . . . . . . . . 10 (Base‘(𝑅 RLocal (RLReg‘𝑅))) = (Base‘(𝑅 RLocal (RLReg‘𝑅)))
132 eqid 2734 . . . . . . . . . 10 (.r‘(𝑅 RLocal (RLReg‘𝑅))) = (.r‘(𝑅 RLocal (RLReg‘𝑅)))
133 eqid 2734 . . . . . . . . . . . 12 (+g𝑅) = (+g𝑅)
13416, 17, 133, 81, 61, 63, 71rloccring 33256 . . . . . . . . . . 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 784 . . . . . . . . . . 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 3974 . . . . . . . . . 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 20272 . . . . . . . . 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 7446 . . . . . . . . . 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 33255 . . . . . . . . . . 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 20272 . . . . . . . . . . . . . 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 20276 . . . . . . . . . . . . . . 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 20286 . . . . . . . . . . . . . 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 20276 . . . . . . . . . . . . . . 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 20285 . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . . 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 20308 . . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . . 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 33262 . . . . . . . . . . . . . . . . . 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 816 . . . . . . . . . . . . . . . . 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 2944 . . . . . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . 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 33287 . . . . . . . . . . . . 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 8796 . . . . . . . . . . 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 2774 . . . . . . . . . 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 2778 . . . . . . . . 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 2774 . . . . . . . 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 4134 . . . . . . . . . 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 2824 . . . . . . . . 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 3974 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘(𝑅 RLocal (RLReg‘𝑅))) ∖ {(0g‘(𝑅 RLocal (RLReg‘𝑅)))})) → 𝑥 ∈ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
184183elrlocbasi 33252 . . . . . 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 3213 . . . . 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 3143 . . . 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 2734 . . . . 5 (0g‘(𝑅 RLocal (RLReg‘𝑅))) = (0g‘(𝑅 RLocal (RLReg‘𝑅)))
188 eqid 2734 . . . . 5 (1r‘(𝑅 RLocal (RLReg‘𝑅))) = (1r‘(𝑅 RLocal (RLReg‘𝑅)))
189 eqid 2734 . . . . 5 (Unit‘(𝑅 RLocal (RLReg‘𝑅))) = (Unit‘(𝑅 RLocal (RLReg‘𝑅)))
190134crngringd 20263 . . . . 5 (𝜑 → (𝑅 RLocal (RLReg‘𝑅)) ∈ Ring)
191131, 187, 188, 132, 189, 190isdrng4 33278 . . . 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 20756 . . 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 2842 1 (𝜑 → ( Frac ‘𝑅) ∈ Field)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  cdif 3959  wss 3962  {csn 4630  cop 4636   class class class wbr 5147   × cxp 5686  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011   Er wer 8740  [cec 8741   / cqs 8742  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  0gc0g 17485  SubMndcsubmnd 18807  Grpcgrp 18963  -gcsg 18965  mulGrpcmgp 20151  1rcur 20198  Ringcrg 20250  CRingccrg 20251  Unitcui 20371  NzRingcnzr 20528  RLRegcrlreg 20707  Domncdomn 20708  IDomncidom 20709  DivRingcdr 20745  Fieldcfield 20746   ~RL cerl 33239   RLocal crloc 33240   Frac cfrac 33283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-ec 8745  df-qs 8749  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-0g 17487  df-imas 17554  df-qus 17555  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-nzr 20529  df-rlreg 20710  df-domn 20711  df-idom 20712  df-drng 20747  df-field 20748  df-erl 33241  df-rloc 33242  df-frac 33284
This theorem is referenced by:  idomsubr  33290  zringfrac  33561
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