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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 7438 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) | |
| 3 | 2 | breq2d 5155 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ (𝑥 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑦))) |
| 4 | breq2 5147 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ 𝑥 ↔ 𝑄 ∥ 𝑋)) | |
| 5 | 4 | orbi1d 917 | . . . 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 7439 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
| 8 | 7 | breq2d 5155 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ (𝑋 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑌))) |
| 9 | breq2 5147 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ 𝑦 ↔ 𝑄 ∥ 𝑌)) | |
| 10 | 9 | orbi2d 916 | . . . 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 2851 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅)) |
| 16 | rprmdvds.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 17 | eqid 2737 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
| 18 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 19 | rprmdvds.d | . . . . . 6 ⊢ ∥ = (∥r‘𝑅) | |
| 20 | rprmdvds.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
| 21 | 16, 17, 18, 19, 20 | isrprm 33545 | . . . . 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 3637 | . 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 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 rprmasso2 33554 rprmirred 33559 rprmdvdspow 33561 rprmdvdsprod 33562 1arithidom 33565 1arithufdlem3 33574 |
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