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Theorem elrlocbasi 33270
Description: Membership in the basis of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypothesis
Ref Expression
elrlocbasi.x (𝜑𝑋 ∈ ((𝐵 × 𝑆) / ))
Assertion
Ref Expression
elrlocbasi (𝜑 → ∃𝑎𝐵𝑏𝑆 𝑋 = [⟨𝑎, 𝑏⟩] )
Distinct variable groups:   ,𝑎,𝑏   𝐵,𝑎,𝑏   𝑆,𝑎,𝑏   𝑋,𝑎,𝑏   𝜑,𝑎,𝑏

Proof of Theorem elrlocbasi
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simp-4r 784 . . . 4 ((((((𝜑𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ) ∧ 𝑎𝐵) ∧ 𝑏𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → 𝑋 = [𝑧] )
2 simpr 484 . . . . 5 ((((((𝜑𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ) ∧ 𝑎𝐵) ∧ 𝑏𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → 𝑧 = ⟨𝑎, 𝑏⟩)
32eceq1d 8785 . . . 4 ((((((𝜑𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ) ∧ 𝑎𝐵) ∧ 𝑏𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → [𝑧] = [⟨𝑎, 𝑏⟩] )
41, 3eqtrd 2777 . . 3 ((((((𝜑𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ) ∧ 𝑎𝐵) ∧ 𝑏𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → 𝑋 = [⟨𝑎, 𝑏⟩] )
5 elxp2 5709 . . . . 5 (𝑧 ∈ (𝐵 × 𝑆) ↔ ∃𝑎𝐵𝑏𝑆 𝑧 = ⟨𝑎, 𝑏⟩)
65biimpi 216 . . . 4 (𝑧 ∈ (𝐵 × 𝑆) → ∃𝑎𝐵𝑏𝑆 𝑧 = ⟨𝑎, 𝑏⟩)
76ad2antlr 727 . . 3 (((𝜑𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ) → ∃𝑎𝐵𝑏𝑆 𝑧 = ⟨𝑎, 𝑏⟩)
84, 7reximddv2 3215 . 2 (((𝜑𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ) → ∃𝑎𝐵𝑏𝑆 𝑋 = [⟨𝑎, 𝑏⟩] )
9 elrlocbasi.x . . 3 (𝜑𝑋 ∈ ((𝐵 × 𝑆) / ))
10 elqsi 8810 . . 3 (𝑋 ∈ ((𝐵 × 𝑆) / ) → ∃𝑧 ∈ (𝐵 × 𝑆)𝑋 = [𝑧] )
119, 10syl 17 . 2 (𝜑 → ∃𝑧 ∈ (𝐵 × 𝑆)𝑋 = [𝑧] )
128, 11r19.29a 3162 1 (𝜑 → ∃𝑎𝐵𝑏𝑆 𝑋 = [⟨𝑎, 𝑏⟩] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  cop 4632   × cxp 5683  [cec 8743   / cqs 8744
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-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751
This theorem is referenced by:  rloccring  33274  rloc1r  33276  fracfld  33310  zringfrac  33582
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