Theorem fracerl 33787

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	⊢ · = (.r‘𝑅)
fracerl.3	⊢ ∼ = (𝑅 ~RL (RLReg‘𝑅))
fracerl.4	⊢ (𝜑 → 𝑅 ∈ CRing)
fracerl.5	⊢ (𝜑 → 𝐸 ∈ 𝐵)
fracerl.6	⊢ (𝜑 → 𝐺 ∈ 𝐵)
fracerl.7	⊢ (𝜑 → 𝐹 ∈ (RLReg‘𝑅))
fracerl.8	⊢ (𝜑 → 𝐻 ∈ (RLReg‘𝑅))

Assertion

Ref	Expression
fracerl	⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))

Proof of Theorem fracerl

Dummy variables 𝑎 𝑏 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fracerl.3	. . . . 5 ⊢ ∼ = (𝑅 ~RL (RLReg‘𝑅))
2		fracerl.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
3		eqid 2760	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
4		fracerl.2	. . . . . 6 ⊢ · = (.r‘𝑅)
5		eqid 2760	. . . . . 6 ⊢ (-g‘𝑅) = (-g‘𝑅)
6		eqid 2760	. . . . . 6 ⊢ (𝐵 × (RLReg‘𝑅)) = (𝐵 × (RLReg‘𝑅))
7		eqid 2760	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (0g‘𝑅))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (0g‘𝑅))}
8		eqid 2760	. . . . . . . 8 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
9	8, 2	rrgss 20901	. . . . . . 7 ⊢ (RLReg‘𝑅) ⊆ 𝐵
10	9	a1i 11	. . . . . 6 ⊢ (𝜑 → (RLReg‘𝑅) ⊆ 𝐵)
11	2, 3, 4, 5, 6, 7, 10	erlval 33738	. . . . 5 ⊢ (𝜑 → (𝑅 ~RL (RLReg‘𝑅)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (0g‘𝑅))})
12	1, 11	eqtrid 2807	. . . 4 ⊢ (𝜑 → ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (0g‘𝑅))})
13		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑎 = ⟨𝐸, 𝐹⟩)
14	13	fveq2d 6878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑎) = (1st ‘⟨𝐸, 𝐹⟩))
15		fracerl.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
16		fracerl.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (RLReg‘𝑅))
17	16	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝐹 ∈ (RLReg‘𝑅))
18		op1stg 7997	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
19	15, 17, 18	syl2an2r 698	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
20	14, 19	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑎) = 𝐸)
21		simprr 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑏 = ⟨𝐺, 𝐻⟩)
22	21	fveq2d 6878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑏) = (2nd ‘⟨𝐺, 𝐻⟩))
23		fracerl.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
24		fracerl.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ (RLReg‘𝑅))
25	24	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝐻 ∈ (RLReg‘𝑅))
26		op2ndg 7998	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
27	23, 25, 26	syl2an2r 698	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
28	22, 27	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑏) = 𝐻)
29	20, 28	oveq12d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
30	21	fveq2d 6878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑏) = (1st ‘⟨𝐺, 𝐻⟩))
31		op1stg 7997	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
32	23, 25, 31	syl2an2r 698	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
33	30, 32	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑏) = 𝐺)
34	13	fveq2d 6878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑎) = (2nd ‘⟨𝐸, 𝐹⟩))
35		op2ndg 7998	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
36	15, 17, 35	syl2an2r 698	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
37	34, 36	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑎) = 𝐹)
38	33, 37	oveq12d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐺 · 𝐹))
39	29, 38	oveq12d 7427	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))) = ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)))
40	39	oveq2d 7425	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))))
41	40	eqeq1d 2762	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (0g‘𝑅) ↔ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)))
42	41	rexbidv 3186	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (0g‘𝑅) ↔ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)))
43	12, 42	brab2d 5509	. . 3 ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ ((⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)) ∧ ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))))
44	15, 16	opelxpd 5687	. . . . 5 ⊢ (𝜑 → ⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)))
45	23, 24	opelxpd 5687	. . . . 5 ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅)))
46	44, 45	jca 521	. . . 4 ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)) ∧ ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅))))
47	46	biantrurd 542	. . 3 ⊢ (𝜑 → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) ↔ ((⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)) ∧ ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))))
48		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑡 ∈ (RLReg‘𝑅))
49		fracerl.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
50	49	crnggrpd 20421	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
51	50	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Grp)
52	49	crngringd 20420	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
53	52	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Ring)
54	15	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐸 ∈ 𝐵)
55	9, 24	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ 𝐵)
56	55	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐻 ∈ 𝐵)
57	2, 4, 53, 54, 56	ringcld 20431	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐸 · 𝐻) ∈ 𝐵)
58	23	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐺 ∈ 𝐵)
59	9, 16	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
60	59	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐹 ∈ 𝐵)
61	2, 4, 53, 58, 60	ringcld 20431	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐺 · 𝐹) ∈ 𝐵)
62	2, 5	grpsubcl 19177	. . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
63	51, 57, 61, 62	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
64		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))
65	8, 2, 4, 3	rrgeq0i 20898	. . . . . . 7 ⊢ ((𝑡 ∈ (RLReg‘𝑅) ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵) → ((𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)))
66	65	imp 412	. . . . . 6 ⊢ (((𝑡 ∈ (RLReg‘𝑅) ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))
67	48, 63, 64, 66	syl21anc 851	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))
68	67	r19.29an 3166	. . . 4 ⊢ ((𝜑 ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))
69		oveq1 7416	. . . . . 6 ⊢ (𝑡 = (1r‘𝑅) → (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))))
70	69	eqeq1d 2762	. . . . 5 ⊢ (𝑡 = (1r‘𝑅) → ((𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) ↔ ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)))
71		eqid 2760	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
72	71, 8, 52	1rrg 33763	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ∈ (RLReg‘𝑅))
73	72	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → (1r‘𝑅) ∈ (RLReg‘𝑅))
74		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))
75	74	oveq2d 7425	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = ((1r‘𝑅) · (0g‘𝑅)))
76	2, 71	ringidcl 20441	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
77	52, 76	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵)
78	2, 4, 3, 52, 77	ringrzd 20474	. . . . . . 7 ⊢ (𝜑 → ((1r‘𝑅) · (0g‘𝑅)) = (0g‘𝑅))
79	78	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((1r‘𝑅) · (0g‘𝑅)) = (0g‘𝑅))
80	75, 79	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))
81	70, 73, 80	rspcedvdw 3579	. . . 4 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))
82	68, 81	impbida 813	. . 3 ⊢ (𝜑 → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) ↔ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)))
83	43, 47, 82	3bitr2d 310	. 2 ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)))
84	2, 4, 52, 15, 55	ringcld 20431	. . 3 ⊢ (𝜑 → (𝐸 · 𝐻) ∈ 𝐵)
85	2, 4, 52, 23, 59	ringcld 20431	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ 𝐵)
86	2, 3, 5	grpsubeq0 19183	. . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → (((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
87	50, 84, 85, 86	syl3anc 1398	. 2 ⊢ (𝜑 → (((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
88	83, 87	bitrd 282	1 ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103  {copab 5167   × cxp 5646  ‘cfv 6528  (class class class)co 7409  1^st c1st 7983  2^nd c2nd 7984  Basecbs 17334  ._rcmulr 17376  0_gc0g 17557  Grpcgrp 19091  -_gcsg 19093  1_rcur 20354  Ringcrg 20406  CRingccrg 20407  RLRegcrlreg 20890   ~_RL cerl 33733

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-cnex 11213  ax-resscn 11214  ax-1cn 11215  ax-icn 11216  ax-addcl 11217  ax-addrcl 11218  ax-mulcl 11219  ax-mulrcl 11220  ax-mulcom 11221  ax-addass 11222  ax-mulass 11223  ax-distr 11224  ax-i2m1 11225  ax-1ne0 11226  ax-1rid 11227  ax-rnegex 11228  ax-rrecex 11229  ax-cnre 11230  ax-pre-lttri 11231  ax-pre-lttrn 11232  ax-pre-ltadd 11233  ax-pre-mulgt0 11234

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7862  df-1st 7985  df-2nd 7986  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11500  df-neg 11501  df-nn 12291  df-2 12360  df-3 12361  df-sets 17289  df-slot 17307  df-ndx 17319  df-base 17335  df-ress 17356  df-plusg 17388  df-mulr 17389  df-0g 17559  df-mgm 18763  df-sgrp 18855  df-mnd 18871  df-grp 19094  df-minusg 19095  df-sbg 19096  df-cmn 19943  df-abl 19944  df-mgp 20308  df-rng 20322  df-ur 20355  df-ring 20408  df-cring 20409  df-oppr 20514  df-dvdsr 20534  df-unit 20535  df-invr 20565  df-rlreg 20893  df-erl 33735

This theorem is used by:  fracfld  33789  zringfrac  34005