Theorem elrlocbasi 33270

Description: Membership in the basis of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypothesis

Ref	Expression
elrlocbasi.x	⊢ (𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ ))

Assertion

Ref	Expression
elrlocbasi	⊢ (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 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.

Step	Hyp	Ref	Expression
1		simp-4r 784	. . . 4 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → 𝑋 = [𝑧] ∼ )
2		simpr 484	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → 𝑧 = ⟨𝑎, 𝑏⟩)
3	2	eceq1d 8785	. . . 4 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → [𝑧] ∼ = [⟨𝑎, 𝑏⟩] ∼ )
4	1, 3	eqtrd 2777	. . 3 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )
5		elxp2 5709	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝑆) ↔ ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑧 = ⟨𝑎, 𝑏⟩)
6	5	biimpi 216	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝑆) → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑧 = ⟨𝑎, 𝑏⟩)
7	6	ad2antlr 727	. . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑧 = ⟨𝑎, 𝑏⟩)
8	4, 7	reximddv2 3215	. 2 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )
9		elrlocbasi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ ))
10		elqsi 8810	. . 3 ⊢ (𝑋 ∈ ((𝐵 × 𝑆) / ∼ ) → ∃𝑧 ∈ (𝐵 × 𝑆)𝑋 = [𝑧] ∼ )
11	9, 10	syl 17	. 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝐵 × 𝑆)𝑋 = [𝑧] ∼ )
12	8, 11	r19.29a 3162	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )

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