Theorem erlcl2 33185

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
erlcl2	⊢ (𝜑 → 𝑉 ∈ (𝐵 × 𝑆))

Proof of Theorem erlcl2

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 2729	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2729	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2729	. . . . . 6 ⊢ (-g‘𝑅) = (-g‘𝑅)
7		eqid 2729	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
8		eqid 2729	. . . . . 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 33182	. . . . 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 2776	. . . 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 482	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑎 = 𝑈)
13	12	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑎) = (1st ‘𝑈))
14		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑏 = 𝑉)
15	14	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑏) = (2nd ‘𝑉))
16	13, 15	oveq12d 7387	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏)) = ((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉)))
17	14	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑏) = (1st ‘𝑉))
18	12	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑎) = (2nd ‘𝑈))
19	17, 18	oveq12d 7387	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)) = ((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))
20	16, 19	oveq12d 7387	. . . . . . . 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 7385	. . . . . . 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 2731	. . . . . 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 3157	. . . . 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 481	. . . 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 32508	. . 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 232	. 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	simplrd 769	1 ⊢ (𝜑 → 𝑉 ∈ (𝐵 × 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3911   class class class wbr 5102  {copab 5164   × cxp 5629  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  ._rcmulr 17197  0_gc0g 17378  -_gcsg 18843   ~_RL cerl 33177

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-erl 33179

This theorem is referenced by:  rlocaddval  33192  rlocmulval  33193