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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 2848 | . 2 ⊢ (𝜑 → 𝑋 ∈ (RPrime‘𝑅)) |
5 | rprmcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | eqid 2734 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
7 | eqid 2734 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | eqid 2734 | . . . . 5 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
9 | eqid 2734 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | 5, 6, 7, 8, 9 | isrprm 33524 | . . . 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 3974 | . 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 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 rprmasso 33532 rprmasso2 33533 rprmasso3 33534 unitmulrprm 33535 rprmirred 33538 1arithidomlem1 33542 1arithidomlem2 33543 1arithidom 33544 1arithufdlem1 33551 1arithufdlem2 33552 1arithufdlem3 33553 1arithufdlem4 33554 dfufd2lem 33556 |
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