| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rprmnunit | Structured version Visualization version GIF version | ||
| Description: A ring prime is not a unit. (Contributed by Thierry Arnoux, 18-May-2025.) |
| Ref | Expression |
|---|---|
| rprmdvds.2 | ⊢ 𝑃 = (RPrime‘𝑅) |
| rprmdvds.3 | ⊢ 𝑈 = (Unit‘𝑅) |
| rprmdvds.5 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
| rprmdvds.6 | ⊢ (𝜑 → 𝑄 ∈ 𝑃) |
| Ref | Expression |
|---|---|
| rprmnunit | ⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqidd 2738 | . 2 ⊢ (𝜑 → (𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)})) | |
| 2 | rprmdvds.5 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
| 3 | rprmdvds.6 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃) | |
| 4 | rprmdvds.2 | . . . . 5 ⊢ 𝑃 = (RPrime‘𝑅) | |
| 5 | 3, 4 | eleqtrdi 2851 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅)) |
| 6 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 7 | rprmdvds.3 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
| 8 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 9 | eqid 2737 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
| 10 | eqid 2737 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 11 | 6, 7, 8, 9, 10 | isrprm 33545 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑄 ∈ (RPrime‘𝑅) ↔ (𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)})) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑄(∥r‘𝑅)(𝑥(.r‘𝑅)𝑦) → (𝑄(∥r‘𝑅)𝑥 ∨ 𝑄(∥r‘𝑅)𝑦))))) |
| 12 | 11 | simprbda 498 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → 𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)}))) |
| 13 | 2, 5, 12 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)}))) |
| 14 | 13 | eldifbd 3964 | . 2 ⊢ (𝜑 → ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)})) |
| 15 | nelun 32532 | . . 3 ⊢ ((𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)}) → (¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)}) ↔ (¬ 𝑄 ∈ 𝑈 ∧ ¬ 𝑄 ∈ {(0g‘𝑅)}))) | |
| 16 | 15 | simprbda 498 | . 2 ⊢ (((𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)}) ∧ ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)})) → ¬ 𝑄 ∈ 𝑈) |
| 17 | 1, 14, 16 | syl2anc 584 | 1 ⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∨ 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 rprmndvdsr1 33552 rprmirred 33559 1arithidom 33565 |
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