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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ufdprmidl | Structured version Visualization version GIF version | ||
| Description: In a unique factorization domain 𝑅, a nonzero prime ideal 𝐽 contains a prime element 𝑝. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| Ref | Expression |
|---|---|
| isufd.i | ⊢ 𝐼 = (PrmIdeal‘𝑅) |
| isufd.3 | ⊢ 𝑃 = (RPrime‘𝑅) |
| isufd.0 | ⊢ 0 = (0g‘𝑅) |
| ufdprmidl.2 | ⊢ (𝜑 → 𝑅 ∈ UFD) |
| ufdprmidl.3 | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
| ufdprmidl.4 | ⊢ (𝜑 → 𝐽 ≠ { 0 }) |
| Ref | Expression |
|---|---|
| ufdprmidl | ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ineq1 4213 | . . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ 𝑃) = (𝐽 ∩ 𝑃)) | |
| 2 | 1 | neeq1d 3000 | . . . 4 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅)) |
| 3 | 2 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅)) |
| 4 | incom 4209 | . . . . 5 ⊢ (𝑃 ∩ 𝐽) = (𝐽 ∩ 𝑃) | |
| 5 | 4 | neeq1i 3005 | . . . 4 ⊢ ((𝑃 ∩ 𝐽) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅) |
| 6 | inn0 4372 | . . . 4 ⊢ ((𝑃 ∩ 𝐽) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽) | |
| 7 | 5, 6 | bitr3i 277 | . . 3 ⊢ ((𝐽 ∩ 𝑃) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽) |
| 8 | 3, 7 | bitrdi 287 | . 2 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)) |
| 9 | ufdprmidl.3 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
| 10 | ufdprmidl.4 | . . 3 ⊢ (𝜑 → 𝐽 ≠ { 0 }) | |
| 11 | 9, 10 | eldifsnd 4787 | . 2 ⊢ (𝜑 → 𝐽 ∈ (𝐼 ∖ {{ 0 }})) |
| 12 | ufdprmidl.2 | . . 3 ⊢ (𝜑 → 𝑅 ∈ UFD) | |
| 13 | isufd.i | . . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅) | |
| 14 | isufd.3 | . . . . 5 ⊢ 𝑃 = (RPrime‘𝑅) | |
| 15 | isufd.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
| 16 | 13, 14, 15 | isufd 33568 | . . . 4 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)) |
| 17 | 16 | simprbi 496 | . . 3 ⊢ (𝑅 ∈ UFD → ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅) |
| 18 | 12, 17 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅) |
| 19 | 8, 11, 18 | rspcdv2 3617 | 1 ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∖ cdif 3948 ∩ cin 3950 ∅c0 4333 {csn 4626 ‘cfv 6561 0gc0g 17484 RPrimecrpm 20432 IDomncidom 20693 PrmIdealcprmidl 33463 UFDcufd 33566 |
| 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-11 2157 ax-12 2177 ax-ext 2708 |
| 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-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-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ufd 33567 |
| This theorem is referenced by: 1arithufdlem1 33572 |
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