![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ufdidom | Structured version Visualization version GIF version |
Description: A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
Ref | Expression |
---|---|
ufdidom.2 | ⊢ (𝜑 → 𝑅 ∈ UFD) |
Ref | Expression |
---|---|
ufdidom | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ufdidom.2 | . 2 ⊢ (𝜑 → 𝑅 ∈ UFD) | |
2 | eqid 2736 | . . . 4 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅) | |
3 | eqid 2736 | . . . 4 ⊢ (RPrime‘𝑅) = (RPrime‘𝑅) | |
4 | eqid 2736 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | 2, 3, 4 | isufd 33555 | . . 3 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{(0g‘𝑅)}})(𝑖 ∩ (RPrime‘𝑅)) ≠ ∅)) |
6 | 5 | simplbi 497 | . 2 ⊢ (𝑅 ∈ UFD → 𝑅 ∈ IDomn) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝜑 → 𝑅 ∈ IDomn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2939 ∀wral 3060 ∖ cdif 3947 ∩ cin 3949 ∅c0 4332 {csn 4624 ‘cfv 6559 0gc0g 17480 RPrimecrpm 20424 IDomncidom 20685 PrmIdealcprmidl 33450 UFDcufd 33553 |
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-ext 2707 |
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-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-iota 6512 df-fv 6567 df-ufd 33554 |
This theorem is referenced by: 1arithufdlem1 33559 1arithufdlem2 33560 1arithufdlem3 33561 1arithufdlem4 33562 dfufd2 33565 |
Copyright terms: Public domain | W3C validator |