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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 2735 | . 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 2848 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅)) |
6 | eqid 2734 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | rprmdvds.3 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
8 | eqid 2734 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | eqid 2734 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
10 | eqid 2734 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 6, 7, 8, 9, 10 | isrprm 33524 | . . . . 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 3975 | . 2 ⊢ (𝜑 → ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)})) |
15 | nelun 32540 | . . 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 847 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∖ cdif 3959 ∪ cun 3960 {csn 4630 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 .rcmulr 17298 0gc0g 17485 ∥rcdsr 20370 Unitcui 20371 RPrimecrpm 20448 |
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-10 2138 ax-11 2154 ax-12 2174 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-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-rprm 20449 |
This theorem is referenced by: rsprprmprmidl 33529 rprmndvdsr1 33531 rprmirred 33538 1arithidom 33544 |
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