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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 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〉] ∼ ) |
| 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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