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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrlocbasi | Structured version Visualization version GIF version |
Description: Membership in the basis of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
Ref | Expression |
---|---|
elrlocbasi.x | ⊢ (𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ )) |
Ref | Expression |
---|---|
elrlocbasi | ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [〈𝑎, 𝑏〉] ∼ ) |
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 8783 | . . . 4 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = 〈𝑎, 𝑏〉) → [𝑧] ∼ = [〈𝑎, 𝑏〉] ∼ ) |
4 | 1, 3 | eqtrd 2774 | . . 3 ⊢ ((((((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝑆) ∧ 𝑧 = 〈𝑎, 𝑏〉) → 𝑋 = [〈𝑎, 𝑏〉] ∼ ) |
5 | elxp2 5712 | . . . . 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 3212 | . 2 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝑆)) ∧ 𝑋 = [𝑧] ∼ ) → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [〈𝑎, 𝑏〉] ∼ ) |
9 | elrlocbasi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ )) | |
10 | elqsi 8808 | . . 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 3159 | 1 ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [〈𝑎, 𝑏〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 〈cop 4636 × cxp 5686 [cec 8741 / cqs 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ec 8745 df-qs 8749 |
This theorem is referenced by: rloccring 33256 rloc1r 33258 fracfld 33289 zringfrac 33561 |
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