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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rprmdvds | Structured version Visualization version GIF version |
Description: If a ring prime 𝑄 divides a product 𝑋 · 𝑌, then it divides either 𝑋 or 𝑌. (Contributed by Thierry Arnoux, 18-May-2025.) |
Ref | Expression |
---|---|
rprmdvds.b | ⊢ 𝐵 = (Base‘𝑅) |
rprmdvds.p | ⊢ 𝑃 = (RPrime‘𝑅) |
rprmdvds.d | ⊢ ∥ = (∥r‘𝑅) |
rprmdvds.t | ⊢ · = (.r‘𝑅) |
rprmdvds.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
rprmdvds.q | ⊢ (𝜑 → 𝑄 ∈ 𝑃) |
rprmdvds.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rprmdvds.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
rprmdvds.1 | ⊢ (𝜑 → 𝑄 ∥ (𝑋 · 𝑌)) |
Ref | Expression |
---|---|
rprmdvds | ⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rprmdvds.1 | . 2 ⊢ (𝜑 → 𝑄 ∥ (𝑋 · 𝑌)) | |
2 | oveq1 7437 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) | |
3 | 2 | breq2d 5159 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ (𝑥 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑦))) |
4 | breq2 5151 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ 𝑥 ↔ 𝑄 ∥ 𝑋)) | |
5 | 4 | orbi1d 916 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦) ↔ (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦))) |
6 | 3, 5 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦)) ↔ (𝑄 ∥ (𝑋 · 𝑦) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦)))) |
7 | oveq2 7438 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
8 | 7 | breq2d 5159 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ (𝑋 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑌))) |
9 | breq2 5151 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ 𝑦 ↔ 𝑄 ∥ 𝑌)) | |
10 | 9 | orbi2d 915 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦) ↔ (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))) |
11 | 8, 10 | imbi12d 344 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑄 ∥ (𝑋 · 𝑦) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦)) ↔ (𝑄 ∥ (𝑋 · 𝑌) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)))) |
12 | rprmdvds.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
13 | rprmdvds.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃) | |
14 | rprmdvds.p | . . . . 5 ⊢ 𝑃 = (RPrime‘𝑅) | |
15 | 13, 14 | eleqtrdi 2848 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅)) |
16 | rprmdvds.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
17 | eqid 2734 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
18 | eqid 2734 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
19 | rprmdvds.d | . . . . . 6 ⊢ ∥ = (∥r‘𝑅) | |
20 | rprmdvds.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
21 | 16, 17, 18, 19, 20 | isrprm 33524 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑄 ∈ (RPrime‘𝑅) ↔ (𝑄 ∈ (𝐵 ∖ ((Unit‘𝑅) ∪ {(0g‘𝑅)})) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦))))) |
22 | 21 | simplbda 499 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦))) |
23 | 12, 15, 22 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦))) |
24 | rprmdvds.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
25 | rprmdvds.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
26 | 6, 11, 23, 24, 25 | rspc2dv 3636 | . 2 ⊢ (𝜑 → (𝑄 ∥ (𝑋 · 𝑌) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))) |
27 | 1, 26 | mpd 15 | 1 ⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → 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 rprmasso2 33533 rprmirred 33538 rprmdvdspow 33540 rprmdvdsprod 33541 1arithidom 33544 1arithufdlem3 33553 |
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