Theorem fracerl 33300

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 2735	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
4		fracerl.2	. . . . . 6 ⊢ · = (.r‘𝑅)
5		eqid 2735	. . . . . 6 ⊢ (-g‘𝑅) = (-g‘𝑅)
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 · (((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 2735	. . . . . . . 8 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
9	8, 2	rrgss 20662	. . . . . . 7 ⊢ (RLReg‘𝑅) ⊆ 𝐵
10	9	a1i 11	. . . . . 6 ⊢ (𝜑 → (RLReg‘𝑅) ⊆ 𝐵)
11	2, 3, 4, 5, 6, 7, 10	erlval 33253	. . . . 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 2782	. . . 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 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑎 = ⟨𝐸, 𝐹⟩)
14	13	fveq2d 6880	. . . . . . . . . 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 8000	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
19	15, 17, 18	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐸, 𝐹⟩) = 𝐸)
20	14, 19	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑎) = 𝐸)
21		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → 𝑏 = ⟨𝐺, 𝐻⟩)
22	21	fveq2d 6880	. . . . . . . . . 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 8001	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
27	23, 25, 26	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐺, 𝐻⟩) = 𝐻)
28	22, 27	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑏) = 𝐻)
29	20, 28	oveq12d 7423	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
30	21	fveq2d 6880	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑏) = (1st ‘⟨𝐺, 𝐻⟩))
31		op1stg 8000	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
32	23, 25, 31	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
33	30, 32	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (1st ‘𝑏) = 𝐺)
34	13	fveq2d 6880	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑎) = (2nd ‘⟨𝐸, 𝐹⟩))
35		op2ndg 8001	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
36	15, 17, 35	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘⟨𝐸, 𝐹⟩) = 𝐹)
37	34, 36	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → (2nd ‘𝑎) = 𝐹)
38	33, 37	oveq12d 7423	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐹⟩ ∧ 𝑏 = ⟨𝐺, 𝐻⟩)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐺 · 𝐹))
39	29, 38	oveq12d 7423	. . . . . . 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 7421	. . . . . 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 2737	. . . . 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 3164	. . . 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 32587	. . 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 5693	. . . . 5 ⊢ (𝜑 → ⟨𝐸, 𝐹⟩ ∈ (𝐵 × (RLReg‘𝑅)))
45	23, 24	opelxpd 5693	. . . . 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 768	. . . . . 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 20207	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
51	50	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Grp)
52	49	crngringd 20206	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
53	52	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Ring)
54	15	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐸 ∈ 𝐵)
55	9, 24	sselid 3956	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ 𝐵)
56	55	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐻 ∈ 𝐵)
57	2, 4, 53, 54, 56	ringcld 20220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐸 · 𝐻) ∈ 𝐵)
58	23	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐺 ∈ 𝐵)
59	9, 16	sselid 3956	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
60	59	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐹 ∈ 𝐵)
61	2, 4, 53, 58, 60	ringcld 20220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐺 · 𝐹) ∈ 𝐵)
62	2, 5	grpsubcl 19003	. . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵)
63	51, 57, 61, 62	syl3anc 1373	. . . . . 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 20659	. . . . . . 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 837	. . . . 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 3144	. . . 4 ⊢ ((𝜑 ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))
69		oveq1 7412	. . . . . 6 ⊢ (𝑡 = (1r‘𝑅) → (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))))
70	69	eqeq1d 2737	. . . . 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 2735	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
72	71, 8, 52	1rrg 33277	. . . . . 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 7421	. . . . . 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 20225	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
77	52, 76	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵)
78	2, 4, 3, 52, 77	ringrzd 20256	. . . . . . 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 2770	. . . . 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 3604	. . . 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 800	. . 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 20220	. . 3 ⊢ (𝜑 → (𝐸 · 𝐻) ∈ 𝐵)
85	2, 4, 52, 23, 59	ringcld 20220	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ 𝐵)
86	2, 3, 5	grpsubeq0 19009	. . 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 1373	. 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 1540   ∈ wcel 2108  ∃wrex 3060   ⊆ wss 3926  ⟨cop 4607   class class class wbr 5119  {copab 5181   × cxp 5652  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17228  ._rcmulr 17272  0_gc0g 17453  Grpcgrp 18916  -_gcsg 18918  1_rcur 20141  Ringcrg 20193  CRingccrg 20194  RLRegcrlreg 20651   ~_RL cerl 33248

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-sbg 18921  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-cring 20196  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-rlreg 20654  df-erl 33250

This theorem is referenced by:  fracfld  33302  zringfrac  33569