Theorem erlbrd 33704

Description: Deduce the ring localization equivalence relation. If for some 𝑇 ∈ 𝑆 we have 𝑇 · (𝐸 · 𝐻 − 𝐹 · 𝐺) = 0, then pairs ⟨𝐸, 𝐺⟩ and ⟨𝐹, 𝐻⟩ are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
erlcl1.b	⊢ 𝐵 = (Base‘𝑅)
erlcl1.e	⊢ ∼ = (𝑅 ~RL 𝑆)
erlcl1.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
erldi.1	⊢ 0 = (0g‘𝑅)
erldi.2	⊢ · = (.r‘𝑅)
erldi.3	⊢ − = (-g‘𝑅)
erlbrd.u	⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)
erlbrd.v	⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)
erlbrd.e	⊢ (𝜑 → 𝐸 ∈ 𝐵)
erlbrd.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
erlbrd.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
erlbrd.h	⊢ (𝜑 → 𝐻 ∈ 𝑆)
erlbrd.1	⊢ (𝜑 → 𝑇 ∈ 𝑆)
erlbrd.2	⊢ (𝜑 → (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )

Assertion

Ref	Expression
erlbrd	⊢ (𝜑 → 𝑈 ∼ 𝑉)

Proof of Theorem erlbrd

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

Step	Hyp	Ref	Expression
1		erlbrd.u	. . . . 5 ⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)
2		erlbrd.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
3		erlbrd.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
4	2, 3	opelxpd 5694	. . . . 5 ⊢ (𝜑 → ⟨𝐸, 𝐺⟩ ∈ (𝐵 × 𝑆))
5	1, 4	eqeltrd 2860	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (𝐵 × 𝑆))
6		erlbrd.v	. . . . 5 ⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)
7		erlbrd.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
8		erlbrd.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑆)
9	7, 8	opelxpd 5694	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝐵 × 𝑆))
10	6, 9	eqeltrd 2860	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (𝐵 × 𝑆))
11	5, 10	jca 521	. . 3 ⊢ (𝜑 → (𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)))
12		erlbrd.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
13		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇)
14	13	oveq1d 7429	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 = 𝑇) → (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))))
15	14	eqeq1d 2762	. . . 4 ⊢ ((𝜑 ∧ 𝑡 = 𝑇) → ((𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ↔ (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
16		erlbrd.2	. . . 4 ⊢ (𝜑 → (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )
17	12, 15, 16	rspcedvd 3578	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )
18	11, 17	jca 521	. 2 ⊢ (𝜑 → ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
19		erlcl1.e	. . . 4 ⊢ ∼ = (𝑅 ~RL 𝑆)
20		erlcl1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
21		erldi.1	. . . . 5 ⊢ 0 = (0g‘𝑅)
22		erldi.2	. . . . 5 ⊢ · = (.r‘𝑅)
23		erldi.3	. . . . 5 ⊢ − = (-g‘𝑅)
24		eqid 2760	. . . . 5 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
25		eqid 2760	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}
26		erlcl1.s	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
27	20, 21, 22, 23, 24, 25, 26	erlval 33699	. . . 4 ⊢ (𝜑 → (𝑅 ~RL 𝑆) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
28	19, 27	eqtrid 2807	. . 3 ⊢ (𝜑 → ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
29		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → 𝑎 = 𝑈)
30	29	fveq2d 6883	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑎) = (1st ‘𝑈))
31	1	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝑈) = (1st ‘⟨𝐸, 𝐺⟩))
32		op1stg 7999	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (1st ‘⟨𝐸, 𝐺⟩) = 𝐸)
33	2, 3, 32	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘⟨𝐸, 𝐺⟩) = 𝐸)
34	31, 33	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑈) = 𝐸)
35	34	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑈) = 𝐸)
36	30, 35	eqtrd 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑎) = 𝐸)
37		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → 𝑏 = 𝑉)
38	37	fveq2d 6883	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑏) = (2nd ‘𝑉))
39	6	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘𝑉) = (2nd ‘⟨𝐹, 𝐻⟩))
40		op2ndg 8000	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (2nd ‘⟨𝐹, 𝐻⟩) = 𝐻)
41	7, 8, 40	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐻⟩) = 𝐻)
42	39, 41	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝑉) = 𝐻)
43	42	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑉) = 𝐻)
44	38, 43	eqtrd 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑏) = 𝐻)
45	36, 44	oveq12d 7432	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
46	37	fveq2d 6883	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑏) = (1st ‘𝑉))
47	6	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝑉) = (1st ‘⟨𝐹, 𝐻⟩))
48		op1stg 7999	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (1st ‘⟨𝐹, 𝐻⟩) = 𝐹)
49	7, 8, 48	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐻⟩) = 𝐹)
50	47, 49	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑉) = 𝐹)
51	50	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑉) = 𝐹)
52	46, 51	eqtrd 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑏) = 𝐹)
53	29	fveq2d 6883	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑎) = (2nd ‘𝑈))
54	1	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘𝑈) = (2nd ‘⟨𝐸, 𝐺⟩))
55		op2ndg 8000	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (2nd ‘⟨𝐸, 𝐺⟩) = 𝐺)
56	2, 3, 55	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘⟨𝐸, 𝐺⟩) = 𝐺)
57	54, 56	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝑈) = 𝐺)
58	57	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑈) = 𝐺)
59	53, 58	eqtrd 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑎) = 𝐺)
60	52, 59	oveq12d 7432	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐹 · 𝐺))
61	45, 60	oveq12d 7432	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = ((𝐸 · 𝐻) − (𝐹 · 𝐺)))
62	61	oveq2d 7430	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))))
63	62	eqeq1d 2762	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
64	63	rexbidv 3186	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
65	28, 64	brab2d 5516	. 2 ⊢ (𝜑 → (𝑈 ∼ 𝑉 ↔ ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )))
66	18, 65	mpbird 260	1 ⊢ (𝜑 → 𝑈 ∼ 𝑉)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103  {copab 5167   × cxp 5653  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17302  ._rcmulr 17344  0_gc0g 17525  -_gcsg 19060   ~_RL cerl 33694

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-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-ral 3077  df-rex 3087  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-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-erl 33696

This theorem is used by:  erlbr2d  33705  erler  33706  rlocaddval  33710  rlocmulval  33711  rloccring  33712