| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rprmcl | Structured version Visualization version GIF version | ||
| Description: A ring prime is an element of the base set. (Contributed by Thierry Arnoux, 18-May-2025.) |
| Ref | Expression |
|---|---|
| rprmcl.b | ⊢ 𝐵 = (Base‘𝑅) |
| rprmcl.p | ⊢ 𝑃 = (RPrime‘𝑅) |
| rprmcl.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
| rprmcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| Ref | Expression |
|---|---|
| rprmcl | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rprmcl.r | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
| 2 | rprmcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 3 | rprmcl.p | . . 3 ⊢ 𝑃 = (RPrime‘𝑅) | |
| 4 | 2, 3 | eleqtrdi 2851 | . 2 ⊢ (𝜑 → 𝑋 ∈ (RPrime‘𝑅)) |
| 5 | rprmcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
| 7 | eqid 2737 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 8 | eqid 2737 | . . . . 5 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 10 | 5, 6, 7, 8, 9 | isrprm 33545 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (RPrime‘𝑅) ↔ (𝑋 ∈ (𝐵 ∖ ((Unit‘𝑅) ∪ {(0g‘𝑅)})) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑋(∥r‘𝑅)(𝑥(.r‘𝑅)𝑦) → (𝑋(∥r‘𝑅)𝑥 ∨ 𝑋(∥r‘𝑅)𝑦))))) |
| 11 | 10 | simprbda 498 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ (RPrime‘𝑅)) → 𝑋 ∈ (𝐵 ∖ ((Unit‘𝑅) ∪ {(0g‘𝑅)}))) |
| 12 | 11 | eldifad 3963 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ (RPrime‘𝑅)) → 𝑋 ∈ 𝐵) |
| 13 | 1, 4, 12 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∪ cun 3949 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 0gc0g 17484 ∥rcdsr 20354 Unitcui 20355 RPrimecrpm 20432 |
| 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-rprm 20433 |
| This theorem is referenced by: rsprprmprmidl 33550 rprmasso 33553 rprmasso2 33554 rprmasso3 33555 unitmulrprm 33556 rprmirred 33559 1arithidomlem1 33563 1arithidomlem2 33564 1arithidom 33565 1arithufdlem1 33572 1arithufdlem2 33573 1arithufdlem3 33574 1arithufdlem4 33575 dfufd2lem 33577 |
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