Theorem fracerl 33390

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 2737	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
4		fracerl.2	. . . . . 6 ⊢ · = (.r‘𝑅)
5		eqid 2737	. . . . . 6 ⊢ (-g‘𝑅) = (-g‘𝑅)
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 · (((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 2737	. . . . . . . 8 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
9	8, 2	rrgss 20639	. . . . . . 7 ⊢ (RLReg‘𝑅) ⊆ 𝐵
10	9	a1i 11	. . . . . 6 ⊢ (𝜑 → (RLReg‘𝑅) ⊆ 𝐵)
11	2, 3, 4, 5, 6, 7, 10	erlval 33342	. . . . 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 2784	. . . 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 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑎 = ⟨𝐸, 𝐹⟩)
14	13	fveq2d 6839	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝐹 ∈ (RLReg‘𝑅))
18		op1stg 7947	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
19	15, 17, 18	syl2an2r 686	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
20	14, 19	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑎) = 𝐸)
21		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑏 = ⟨𝐺, 𝐻⟩)
22	21	fveq2d 6839	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝐻 ∈ (RLReg‘𝑅))
26		op2ndg 7948	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
27	23, 25, 26	syl2an2r 686	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
28	22, 27	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑏) = 𝐻)
29	20, 28	oveq12d 7378	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
30	21	fveq2d 6839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑏) = (1st ‘⟨𝐺, 𝐻⟩))
31		op1stg 7947	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
32	23, 25, 31	syl2an2r 686	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
33	30, 32	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑏) = 𝐺)
34	13	fveq2d 6839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑎) = (2nd ‘⟨𝐸, 𝐹⟩))
35		op2ndg 7948	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
36	15, 17, 35	syl2an2r 686	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
37	34, 36	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑎) = 𝐹)
38	33, 37	oveq12d 7378	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐺 · 𝐹))
39	29, 38	oveq12d 7378	. . . . . . 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 7376	. . . . . 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 2739	. . . . 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 3161	. . . 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 32686	. . 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 5664	. . . . 5 ⊢ (𝜑 → ⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)))
45	23, 24	opelxpd 5664	. . . . 5 ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅)))
46	44, 45	jca 511	. . . 4 ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)) ∧ ⟨𝐺, 𝐻⟩ ∈ (𝐵 × (RLReg‘𝑅))))
47	46	biantrurd 532	. . 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 769	. . . . . 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 20186	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
51	50	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Grp)
52	49	crngringd 20185	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
53	52	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Ring)
54	15	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐸 ∈ 𝐵)
55	9, 24	sselid 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ 𝐵)
56	55	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐻 ∈ 𝐵)
57	2, 4, 53, 54, 56	ringcld 20199	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐸 · 𝐻) ∈ 𝐵)
58	23	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐺 ∈ 𝐵)
59	9, 16	sselid 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
60	59	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐹 ∈ 𝐵)
61	2, 4, 53, 58, 60	ringcld 20199	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐺 · 𝐹) ∈ 𝐵)
62	2, 5	grpsubcl 18954	. . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
63	51, 57, 61, 62	syl3anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
64		simpr 484	. . . . . 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 20636	. . . . . . 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 406	. . . . . 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 838	. . . . 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 3141	. . . 4 ⊢ ((𝜑 ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))
69		oveq1 7367	. . . . . 6 ⊢ (𝑡 = (1r‘𝑅) → (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))))
70	69	eqeq1d 2739	. . . . 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 2737	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
72	71, 8, 52	1rrg 33367	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ∈ (RLReg‘𝑅))
73	72	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → (1r‘𝑅) ∈ (RLReg‘𝑅))
74		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))
75	74	oveq2d 7376	. . . . . 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 20204	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
77	52, 76	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵)
78	2, 4, 3, 52, 77	ringrzd 20235	. . . . . . 7 ⊢ (𝜑 → ((1r‘𝑅) · (0g‘𝑅)) = (0g‘𝑅))
79	78	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((1r‘𝑅) · (0g‘𝑅)) = (0g‘𝑅))
80	75, 79	eqtrd 2772	. . . . 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 3580	. . . 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 801	. . 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 307	. 2 ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)))
84	2, 4, 52, 15, 55	ringcld 20199	. . 3 ⊢ (𝜑 → (𝐸 · 𝐻) ∈ 𝐵)
85	2, 4, 52, 23, 59	ringcld 20199	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ 𝐵)
86	2, 3, 5	grpsubeq0 18960	. . 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 1374	. 2 ⊢ (𝜑 → (((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
88	83, 87	bitrd 279	1 ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3902  ⟨cop 4587   class class class wbr 5099  {copab 5161   × cxp 5623  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17140  ._rcmulr 17182  0_gc0g 17363  Grpcgrp 18867  -_gcsg 18869  1_rcur 20120  Ringcrg 20172  CRingccrg 20173  RLRegcrlreg 20628   ~_RL cerl 33337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18870  df-minusg 18871  df-sbg 18872  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-oppr 20277  df-dvdsr 20297  df-unit 20298  df-invr 20328  df-rlreg 20631  df-erl 33339

This theorem is referenced by:  fracfld  33392  zringfrac  33637