Theorem erlcl1 33340

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 2737	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2737	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2737	. . . . . 6 ⊢ (-g‘𝑅) = (-g‘𝑅)
7		eqid 2737	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
8		eqid 2737	. . . . . 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 33338	. . . . 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 2784	. . . 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 6840	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑎) = (1st ‘𝑈))
14		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑏 = 𝑉)
15	14	fveq2d 6840	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑏) = (2nd ‘𝑉))
16	13, 15	oveq12d 7380	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑎)(.r‘𝑅)(2nd ‘𝑏)) = ((1st ‘𝑈)(.r‘𝑅)(2nd ‘𝑉)))
17	14	fveq2d 6840	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑏) = (1st ‘𝑉))
18	12	fveq2d 6840	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑎) = (2nd ‘𝑈))
19	17, 18	oveq12d 7380	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑏)(.r‘𝑅)(2nd ‘𝑎)) = ((1st ‘𝑉)(.r‘𝑅)(2nd ‘𝑈)))
20	16, 19	oveq12d 7380	. . . . . . . 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 7378	. . . . . . 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 2739	. . . . . 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 3162	. . . . 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 32697	. . 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	simplld 768	1 ⊢ (𝜑 → 𝑈 ∈ (𝐵 × 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890   class class class wbr 5086  {copab 5148   × cxp 5624  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  Basecbs 17174  ._rcmulr 17216  0_gc0g 17397  -_gcsg 18906   ~_RL cerl 33333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-erl 33335

This theorem is referenced by:  rlocaddval  33348  rlocmulval  33349