Theorem elrlocbasi 33190

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 783	. . . 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 8688	. . . 4 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → [𝑧] ∼ = [⟨𝑎, 𝑏⟩] ∼ )
4	1, 3	eqtrd 2764	. . 3 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = ⟨𝑎, 𝑏⟩) → 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )
5		elxp2 5655	. . . . 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 3194	. 2 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )
9		elrlocbasi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ ))
10		elqsi 8716	. . 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 3141	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ⟨cop 4591   × cxp 5629  [cec 8646   / cqs 8647

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-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-qs 8654

This theorem is referenced by:  rloccring  33194  rloc1r  33196  fracfld  33231  zringfrac  33498