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Theorem fracerl 33308
Description: Rewrite the ring localization equivalence relation in the case of a field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)
Hypotheses
Ref Expression
fracerl.1 𝐵 = (Base‘𝑅)
fracerl.2 · = (.r𝑅)
fracerl.3 = (𝑅 ~RL (RLReg‘𝑅))
fracerl.4 (𝜑𝑅 ∈ CRing)
fracerl.5 (𝜑𝐸𝐵)
fracerl.6 (𝜑𝐺𝐵)
fracerl.7 (𝜑𝐹 ∈ (RLReg‘𝑅))
fracerl.8 (𝜑𝐻 ∈ (RLReg‘𝑅))
Assertion
Ref Expression
fracerl (𝜑 → (⟨𝐸, 𝐹𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))

Proof of Theorem fracerl
Dummy variables 𝑎 𝑏 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fracerl.3 . . . . 5 = (𝑅 ~RL (RLReg‘𝑅))
2 fracerl.1 . . . . . 6 𝐵 = (Base‘𝑅)
3 eqid 2737 . . . . . 6 (0g𝑅) = (0g𝑅)
4 fracerl.2 . . . . . 6 · = (.r𝑅)
5 eqid 2737 . . . . . 6 (-g𝑅) = (-g𝑅)
6 eqid 2737 . . . . . 6 (𝐵 × (RLReg‘𝑅)) = (𝐵 × (RLReg‘𝑅))
7 eqid 2737 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (0g𝑅))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (0g𝑅))}
8 eqid 2737 . . . . . . . 8 (RLReg‘𝑅) = (RLReg‘𝑅)
98, 2rrgss 20702 . . . . . . 7 (RLReg‘𝑅) ⊆ 𝐵
109a1i 11 . . . . . 6 (𝜑 → (RLReg‘𝑅) ⊆ 𝐵)
112, 3, 4, 5, 6, 7, 10erlval 33262 . . . . 5 (𝜑 → (𝑅 ~RL (RLReg‘𝑅)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (0g𝑅))})
121, 11eqtrid 2789 . . . 4 (𝜑 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (0g𝑅))})
13 simprl 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑎 = ⟨𝐸, 𝐹⟩)
1413fveq2d 6910 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st𝑎) = (1st ‘⟨𝐸, 𝐹⟩))
15 fracerl.5 . . . . . . . . . . 11 (𝜑𝐸𝐵)
16 fracerl.7 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (RLReg‘𝑅))
1716adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝐹 ∈ (RLReg‘𝑅))
18 op1stg 8026 . . . . . . . . . . 11 ((𝐸𝐵𝐹 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
1915, 17, 18syl2an2r 685 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
2014, 19eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st𝑎) = 𝐸)
21 simprr 773 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑏 = ⟨𝐺, 𝐻⟩)
2221fveq2d 6910 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd𝑏) = (2nd ‘⟨𝐺, 𝐻⟩))
23 fracerl.6 . . . . . . . . . . 11 (𝜑𝐺𝐵)
24 fracerl.8 . . . . . . . . . . . 12 (𝜑𝐻 ∈ (RLReg‘𝑅))
2524adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝐻 ∈ (RLReg‘𝑅))
26 op2ndg 8027 . . . . . . . . . . 11 ((𝐺𝐵𝐻 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
2723, 25, 26syl2an2r 685 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
2822, 27eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd𝑏) = 𝐻)
2920, 28oveq12d 7449 . . . . . . . 8 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st𝑎) · (2nd𝑏)) = (𝐸 · 𝐻))
3021fveq2d 6910 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st𝑏) = (1st ‘⟨𝐺, 𝐻⟩))
31 op1stg 8026 . . . . . . . . . . 11 ((𝐺𝐵𝐻 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
3223, 25, 31syl2an2r 685 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
3330, 32eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st𝑏) = 𝐺)
3413fveq2d 6910 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd𝑎) = (2nd ‘⟨𝐸, 𝐹⟩))
35 op2ndg 8027 . . . . . . . . . . 11 ((𝐸𝐵𝐹 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
3615, 17, 35syl2an2r 685 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
3734, 36eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd𝑎) = 𝐹)
3833, 37oveq12d 7449 . . . . . . . 8 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st𝑏) · (2nd𝑎)) = (𝐺 · 𝐹))
3929, 38oveq12d 7449 . . . . . . 7 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎))) = ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)))
4039oveq2d 7447 . . . . . 6 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))))
4140eqeq1d 2739 . . . . 5 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (0g𝑅) ↔ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)))
4241rexbidv 3179 . . . 4 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (0g𝑅) ↔ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)))
4312, 42brab2d 32619 . . 3 (𝜑 → (⟨𝐸, 𝐹𝐺, 𝐻⟩ ↔ ((⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)) ∧ ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅))))
4415, 16opelxpd 5724 . . . . 5 (𝜑 → ⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)))
4523, 24opelxpd 5724 . . . . 5 (𝜑 → ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅)))
4644, 45jca 511 . . . 4 (𝜑 → (⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)) ∧ ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅))))
4746biantrurd 532 . . 3 (𝜑 → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅) ↔ ((⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)) ∧ ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅))))
48 simplr 769 . . . . . 6 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝑡 ∈ (RLReg‘𝑅))
49 fracerl.4 . . . . . . . . 9 (𝜑𝑅 ∈ CRing)
5049crnggrpd 20244 . . . . . . . 8 (𝜑𝑅 ∈ Grp)
5150ad2antrr 726 . . . . . . 7 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝑅 ∈ Grp)
5249crngringd 20243 . . . . . . . . 9 (𝜑𝑅 ∈ Ring)
5352ad2antrr 726 . . . . . . . 8 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝑅 ∈ Ring)
5415ad2antrr 726 . . . . . . . 8 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝐸𝐵)
559, 24sselid 3981 . . . . . . . . 9 (𝜑𝐻𝐵)
5655ad2antrr 726 . . . . . . . 8 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝐻𝐵)
572, 4, 53, 54, 56ringcld 20257 . . . . . . 7 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → (𝐸 · 𝐻) ∈ 𝐵)
5823ad2antrr 726 . . . . . . . 8 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝐺𝐵)
599, 16sselid 3981 . . . . . . . . 9 (𝜑𝐹𝐵)
6059ad2antrr 726 . . . . . . . 8 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝐹𝐵)
612, 4, 53, 58, 60ringcld 20257 . . . . . . 7 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → (𝐺 · 𝐹) ∈ 𝐵)
622, 5grpsubcl 19038 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
6351, 57, 61, 62syl3anc 1373 . . . . . 6 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
64 simpr 484 . . . . . 6 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅))
658, 2, 4, 3rrgeq0i 20699 . . . . . . 7 ((𝑡 ∈ (RLReg‘𝑅) ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) ∈ 𝐵) → ((𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)))
6665imp 406 . . . . . 6 (((𝑡 ∈ (RLReg‘𝑅) ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) ∈ 𝐵) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅))
6748, 63, 64, 66syl21anc 838 . . . . 5 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅))
6867r19.29an 3158 . . . 4 ((𝜑 ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅))
69 oveq1 7438 . . . . . 6 (𝑡 = (1r𝑅) → (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = ((1r𝑅) · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))))
7069eqeq1d 2739 . . . . 5 (𝑡 = (1r𝑅) → ((𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅) ↔ ((1r𝑅) · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)))
71 eqid 2737 . . . . . . 7 (1r𝑅) = (1r𝑅)
7271, 8, 521rrg 33286 . . . . . 6 (𝜑 → (1r𝑅) ∈ (RLReg‘𝑅))
7372adantr 480 . . . . 5 ((𝜑 ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)) → (1r𝑅) ∈ (RLReg‘𝑅))
74 simpr 484 . . . . . . 7 ((𝜑 ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅))
7574oveq2d 7447 . . . . . 6 ((𝜑 ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)) → ((1r𝑅) · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = ((1r𝑅) · (0g𝑅)))
762, 71ringidcl 20262 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝐵)
7752, 76syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ 𝐵)
782, 4, 3, 52, 77ringrzd 20293 . . . . . . 7 (𝜑 → ((1r𝑅) · (0g𝑅)) = (0g𝑅))
7978adantr 480 . . . . . 6 ((𝜑 ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)) → ((1r𝑅) · (0g𝑅)) = (0g𝑅))
8075, 79eqtrd 2777 . . . . 5 ((𝜑 ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)) → ((1r𝑅) · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅))
8170, 73, 80rspcedvdw 3625 . . . 4 ((𝜑 ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)) → ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅))
8268, 81impbida 801 . . 3 (𝜑 → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅) ↔ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)))
8343, 47, 823bitr2d 307 . 2 (𝜑 → (⟨𝐸, 𝐹𝐺, 𝐻⟩ ↔ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)))
842, 4, 52, 15, 55ringcld 20257 . . 3 (𝜑 → (𝐸 · 𝐻) ∈ 𝐵)
852, 4, 52, 23, 59ringcld 20257 . . 3 (𝜑 → (𝐺 · 𝐹) ∈ 𝐵)
862, 3, 5grpsubeq0 19044 . . 3 ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → (((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
8750, 84, 85, 86syl3anc 1373 . 2 (𝜑 → (((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
8883, 87bitrd 279 1 (𝜑 → (⟨𝐸, 𝐹𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  wss 3951  cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  .rcmulr 17298  0gc0g 17484  Grpcgrp 18951  -gcsg 18953  1rcur 20178  Ringcrg 20230  CRingccrg 20231  RLRegcrlreg 20691   ~RL cerl 33257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-rlreg 20694  df-erl 33259
This theorem is referenced by:  fracfld  33310  zringfrac  33582
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