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Theorem fracerl 33288
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 2735 . . . . . 6 (0g𝑅) = (0g𝑅)
4 fracerl.2 . . . . . 6 · = (.r𝑅)
5 eqid 2735 . . . . . 6 (-g𝑅) = (-g𝑅)
6 eqid 2735 . . . . . 6 (𝐵 × (RLReg‘𝑅)) = (𝐵 × (RLReg‘𝑅))
7 eqid 2735 . . . . . 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 2735 . . . . . . . 8 (RLReg‘𝑅) = (RLReg‘𝑅)
98, 2rrgss 20719 . . . . . . 7 (RLReg‘𝑅) ⊆ 𝐵
109a1i 11 . . . . . 6 (𝜑 → (RLReg‘𝑅) ⊆ 𝐵)
112, 3, 4, 5, 6, 7, 10erlval 33245 . . . . 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 2787 . . . 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 6911 . . . . . . . . . 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 8025 . . . . . . . . . . 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 2775 . . . . . . . . 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 6911 . . . . . . . . . 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 8026 . . . . . . . . . . 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 2775 . . . . . . . . 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 6911 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st𝑏) = (1st ‘⟨𝐺, 𝐻⟩))
31 op1stg 8025 . . . . . . . . . . 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 2775 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st𝑏) = 𝐺)
3413fveq2d 6911 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd𝑎) = (2nd ‘⟨𝐸, 𝐹⟩))
35 op2ndg 8026 . . . . . . . . . . 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 2775 . . . . . . . . 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 2737 . . . . 5 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((𝑡 · (((1st𝑎) · (2nd𝑏))(-g𝑅)((1st𝑏) · (2nd𝑎)))) = (0g𝑅) ↔ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)))
4241rexbidv 3177 . . . 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 32627 . . 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 5728 . . . . 5 (𝜑 → ⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)))
4523, 24opelxpd 5728 . . . . 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 20265 . . . . . . . 8 (𝜑𝑅 ∈ Grp)
5150ad2antrr 726 . . . . . . 7 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝑅 ∈ Grp)
5249crngringd 20264 . . . . . . . . 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 3993 . . . . . . . . 9 (𝜑𝐻𝐵)
5655ad2antrr 726 . . . . . . . 8 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝐻𝐵)
572, 4, 53, 54, 56ringcld 20277 . . . . . . 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 3993 . . . . . . . . 9 (𝜑𝐹𝐵)
6059ad2antrr 726 . . . . . . . 8 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → 𝐹𝐵)
612, 4, 53, 58, 60ringcld 20277 . . . . . . 7 (((𝜑𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)) → (𝐺 · 𝐹) ∈ 𝐵)
622, 5grpsubcl 19051 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
6351, 57, 61, 62syl3anc 1370 . . . . . 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 20716 . . . . . . 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 3156 . . . 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 2737 . . . . 5 (𝑡 = (1r𝑅) → ((𝑡 · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅) ↔ ((1r𝑅) · ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹))) = (0g𝑅)))
71 eqid 2735 . . . . . . 7 (1r𝑅) = (1r𝑅)
7271, 8, 521rrg 33267 . . . . . 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 20280 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝐵)
7752, 76syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ 𝐵)
782, 4, 3, 52, 77ringrzd 20310 . . . . . . 7 (𝜑 → ((1r𝑅) · (0g𝑅)) = (0g𝑅))
7978adantr 480 . . . . . 6 ((𝜑 ∧ ((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅)) → ((1r𝑅) · (0g𝑅)) = (0g𝑅))
8075, 79eqtrd 2775 . . . . 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 20277 . . 3 (𝜑 → (𝐸 · 𝐻) ∈ 𝐵)
852, 4, 52, 23, 59ringcld 20277 . . 3 (𝜑 → (𝐺 · 𝐹) ∈ 𝐵)
862, 3, 5grpsubeq0 19057 . . 3 ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → (((𝐸 · 𝐻)(-g𝑅)(𝐺 · 𝐹)) = (0g𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
8750, 84, 85, 86syl3anc 1370 . 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 1537  wcel 2106  wrex 3068  wss 3963  cop 4637   class class class wbr 5148  {copab 5210   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  .rcmulr 17299  0gc0g 17486  Grpcgrp 18964  -gcsg 18966  1rcur 20199  Ringcrg 20251  CRingccrg 20252  RLRegcrlreg 20708   ~RL cerl 33240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-rlreg 20711  df-erl 33242
This theorem is referenced by:  fracfld  33290  zringfrac  33562
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