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Theorem erlcl2 33265
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 (𝜑𝑆𝐵)
erlcl1.1 (𝜑𝑈 𝑉)
Assertion
Ref Expression
erlcl2 (𝜑𝑉 ∈ (𝐵 × 𝑆))

Proof of Theorem erlcl2
Dummy variables 𝑎 𝑏 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erlcl1.1 . . 3 (𝜑𝑈 𝑉)
2 erlcl1.e . . . . 5 = (𝑅 ~RL 𝑆)
3 erlcl1.b . . . . . 6 𝐵 = (Base‘𝑅)
4 eqid 2737 . . . . . 6 (0g𝑅) = (0g𝑅)
5 eqid 2737 . . . . . 6 (.r𝑅) = (.r𝑅)
6 eqid 2737 . . . . . 6 (-g𝑅) = (-g𝑅)
7 eqid 2737 . . . . . 6 (𝐵 × 𝑆) = (𝐵 × 𝑆)
8 eqid 2737 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (0g𝑅))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (0g𝑅))}
9 erlcl1.s . . . . . 6 (𝜑𝑆𝐵)
103, 4, 5, 6, 7, 8, 9erlval 33262 . . . . 5 (𝜑 → (𝑅 ~RL 𝑆) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (0g𝑅))})
112, 10eqtrid 2789 . . . 4 (𝜑 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (0g𝑅))})
12 simpl 482 . . . . . . . . . . 11 ((𝑎 = 𝑈𝑏 = 𝑉) → 𝑎 = 𝑈)
1312fveq2d 6910 . . . . . . . . . 10 ((𝑎 = 𝑈𝑏 = 𝑉) → (1st𝑎) = (1st𝑈))
14 simpr 484 . . . . . . . . . . 11 ((𝑎 = 𝑈𝑏 = 𝑉) → 𝑏 = 𝑉)
1514fveq2d 6910 . . . . . . . . . 10 ((𝑎 = 𝑈𝑏 = 𝑉) → (2nd𝑏) = (2nd𝑉))
1613, 15oveq12d 7449 . . . . . . . . 9 ((𝑎 = 𝑈𝑏 = 𝑉) → ((1st𝑎)(.r𝑅)(2nd𝑏)) = ((1st𝑈)(.r𝑅)(2nd𝑉)))
1714fveq2d 6910 . . . . . . . . . 10 ((𝑎 = 𝑈𝑏 = 𝑉) → (1st𝑏) = (1st𝑉))
1812fveq2d 6910 . . . . . . . . . 10 ((𝑎 = 𝑈𝑏 = 𝑉) → (2nd𝑎) = (2nd𝑈))
1917, 18oveq12d 7449 . . . . . . . . 9 ((𝑎 = 𝑈𝑏 = 𝑉) → ((1st𝑏)(.r𝑅)(2nd𝑎)) = ((1st𝑉)(.r𝑅)(2nd𝑈)))
2016, 19oveq12d 7449 . . . . . . . 8 ((𝑎 = 𝑈𝑏 = 𝑉) → (((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎))) = (((1st𝑈)(.r𝑅)(2nd𝑉))(-g𝑅)((1st𝑉)(.r𝑅)(2nd𝑈))))
2120oveq2d 7447 . . . . . . 7 ((𝑎 = 𝑈𝑏 = 𝑉) → (𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (𝑡(.r𝑅)(((1st𝑈)(.r𝑅)(2nd𝑉))(-g𝑅)((1st𝑉)(.r𝑅)(2nd𝑈)))))
2221eqeq1d 2739 . . . . . 6 ((𝑎 = 𝑈𝑏 = 𝑉) → ((𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (0g𝑅) ↔ (𝑡(.r𝑅)(((1st𝑈)(.r𝑅)(2nd𝑉))(-g𝑅)((1st𝑉)(.r𝑅)(2nd𝑈)))) = (0g𝑅)))
2322rexbidv 3179 . . . . 5 ((𝑎 = 𝑈𝑏 = 𝑉) → (∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (0g𝑅) ↔ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑈)(.r𝑅)(2nd𝑉))(-g𝑅)((1st𝑉)(.r𝑅)(2nd𝑈)))) = (0g𝑅)))
2423adantl 481 . . . 4 ((𝜑 ∧ (𝑎 = 𝑈𝑏 = 𝑉)) → (∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑎)(.r𝑅)(2nd𝑏))(-g𝑅)((1st𝑏)(.r𝑅)(2nd𝑎)))) = (0g𝑅) ↔ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑈)(.r𝑅)(2nd𝑉))(-g𝑅)((1st𝑉)(.r𝑅)(2nd𝑈)))) = (0g𝑅)))
2511, 24brab2d 32619 . . 3 (𝜑 → (𝑈 𝑉 ↔ ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑈)(.r𝑅)(2nd𝑉))(-g𝑅)((1st𝑉)(.r𝑅)(2nd𝑈)))) = (0g𝑅))))
261, 25mpbid 232 . 2 (𝜑 → ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡𝑆 (𝑡(.r𝑅)(((1st𝑈)(.r𝑅)(2nd𝑉))(-g𝑅)((1st𝑉)(.r𝑅)(2nd𝑈)))) = (0g𝑅)))
2726simplrd 770 1 (𝜑𝑉 ∈ (𝐵 × 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  wss 3951   class class class wbr 5143  {copab 5205   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  .rcmulr 17298  0gc0g 17484  -gcsg 18953   ~RL cerl 33257
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-erl 33259
This theorem is referenced by:  rlocaddval  33272  rlocmulval  33273
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