Theorem erlcl1 33685

Description: Closure for the ring localization equivalence 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)
erlcl1.1	⊢ (𝜑 → 𝑈 ∼ 𝑉)

Assertion

Ref	Expression
erlcl1	⊢ (𝜑 → 𝑈 ∈ (𝐵 × 𝑆))

Proof of Theorem erlcl1

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

Step	Hyp	Ref	Expression
1		erlcl1.1	. . 3 ⊢ (𝜑 → 𝑈 ∼ 𝑉)
2		erlcl1.e	. . . . 5 ⊢ ∼ = (𝑅 ~RL 𝑆)
3		erlcl1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2762	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2762	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2762	. . . . . 6 ⊢ (-g‘𝑅) = (-g‘𝑅)
7		eqid 2762	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
8		eqid 2762	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (0g‘𝑅))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (0g‘𝑅))}
9		erlcl1.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
10	3, 4, 5, 6, 7, 8, 9	erlval 33683	. . . . 5 ⊢ (𝜑 → (𝑅 ~RL 𝑆) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (0g‘𝑅))})
11	2, 10	eqtrid 2809	. . . 4 ⊢ (𝜑 → ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (0g‘𝑅))})
12		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑎 = 𝑈)
13	12	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑎) = (1st ‘𝑈))
14		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑏 = 𝑉)
15	14	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑏) = (2nd ‘𝑉))
16	13, 15	oveq12d 7434	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏)) = ((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉)))
17	14	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑏) = (1st ‘𝑉))
18	12	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑎) = (2nd ‘𝑈))
19	17, 18	oveq12d 7434	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)) = ((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))
20	16, 19	oveq12d 7434	. . . . . . . 8 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎))) = (((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉))(-g‘𝑅)((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈))))
21	20	oveq2d 7432	. . . . . . 7 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (𝑡(.r‘𝑅)(((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉))(-g‘𝑅)((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))))
22	21	eqeq1d 2764	. . . . . 6 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (0g‘𝑅) ↔ (𝑡(.r‘𝑅)(((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉))(-g‘𝑅)((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))) = (0g‘𝑅)))
23	22	rexbidv 3188	. . . . 5 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (0g‘𝑅) ↔ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉))(-g‘𝑅)((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))) = (0g‘𝑅)))
24	23	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)))) = (0g‘𝑅) ↔ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉))(-g‘𝑅)((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))) = (0g‘𝑅)))
25	11, 24	brab2d 5520	. . 3 ⊢ (𝜑 → (𝑈 ∼ 𝑉 ↔ ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉))(-g‘𝑅)((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))) = (0g‘𝑅))))
26	1, 25	mpbid 235	. 2 ⊢ (𝜑 → ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡(.r‘𝑅)(((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉))(-g‘𝑅)((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))) = (0g‘𝑅)))
27	26	simplld 780	1 ⊢ (𝜑 → 𝑈 ∈ (𝐵 × 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902   class class class wbr 5107  {copab 5171   × cxp 5657  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988  Basecbs 17303  ._rcmulr 17345  0_gc0g 17526  -_gcsg 19058   ~_RL cerl 33678

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-erl 33680

This theorem is used by:  rlocaddval  33694  rlocmulval  33695