Theorem erlbrd 33923

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 5697	. . . . 5 ⊢ (𝜑 → ⟨𝐸, 𝐺⟩ ∈ (𝐵 × 𝑆))
5	1, 4	eqeltrd 2864	. . . 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 5697	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝐵 × 𝑆))
10	6, 9	eqeltrd 2864	. . . 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 7466	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 = 𝑇) → (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))))
15	14	eqeq1d 2765	. . . 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 3582	. . 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 2763	. . . . 5 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
25		eqid 2763	. . . . 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 33918	. . . 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 2810	. . 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 6920	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑎) = (1st ‘𝑈))
31	1	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝑈) = (1st ‘⟨𝐸, 𝐺⟩))
32		op1stg 8047	. . . . . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑈) = 𝐸)
35	34	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑈) = 𝐸)
36	30, 35	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑎) = 𝐸)
37		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → 𝑏 = 𝑉)
38	37	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑏) = (2nd ‘𝑉))
39	6	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘𝑉) = (2nd ‘⟨𝐹, 𝐻⟩))
40		op2ndg 8048	. . . . . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝑉) = 𝐻)
43	42	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑉) = 𝐻)
44	38, 43	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑏) = 𝐻)
45	36, 44	oveq12d 7469	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
46	37	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑏) = (1st ‘𝑉))
47	6	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝑉) = (1st ‘⟨𝐹, 𝐻⟩))
48		op1stg 8047	. . . . . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑉) = 𝐹)
51	50	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑉) = 𝐹)
52	46, 51	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑏) = 𝐹)
53	29	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑎) = (2nd ‘𝑈))
54	1	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘𝑈) = (2nd ‘⟨𝐸, 𝐺⟩))
55		op2ndg 8048	. . . . . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝑈) = 𝐺)
58	57	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑈) = 𝐺)
59	53, 58	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑎) = 𝐺)
60	52, 59	oveq12d 7469	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐹 · 𝐺))
61	45, 60	oveq12d 7469	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = ((𝐸 · 𝐻) − (𝐹 · 𝐺)))
62	61	oveq2d 7467	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))))
63	62	eqeq1d 2765	. . . 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 3190	. . 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 5519	. 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 2147  ∃wrex 3090   ⊆ wss 3900  ⟨cop 4840   class class class wbr 5108  {copab 5173   × cxp 5656  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  Basecbs 17431  ._rcmulr 17473  0_gc0g 17660  -_gcsg 19196   ~_RL cerl 33913

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-erl 33915

This theorem is used by:  erlbr2d  33924  erler  33925  rlocaddval  33929  rlocmulval  33930  rloccring  33931