Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isufd | Structured version Visualization version GIF version |
Description: The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
Ref | Expression |
---|---|
isufd.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
isufd.i | ⊢ 𝐼 = (PrmIdeal‘𝑅) |
isufd.3 | ⊢ 𝑃 = (RPrime‘𝑅) |
Ref | Expression |
---|---|
isufd | ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6774 | . . . . 5 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
2 | isufd.a | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
3 | 1, 2 | eqtr4di 2796 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) |
4 | 3 | neeq1d 3003 | . . 3 ⊢ (𝑟 = 𝑅 → ((AbsVal‘𝑟) ≠ ∅ ↔ 𝐴 ≠ ∅)) |
5 | fveq2 6774 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅)) | |
6 | isufd.i | . . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅) | |
7 | 5, 6 | eqtr4di 2796 | . . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼) |
8 | fveq2 6774 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅)) | |
9 | isufd.3 | . . . . . . 7 ⊢ 𝑃 = (RPrime‘𝑅) | |
10 | 8, 9 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃) |
11 | 10 | ineq2d 4146 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖 ∩ 𝑃)) |
12 | 11 | neeq1d 3003 | . . . 4 ⊢ (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅)) |
13 | 7, 12 | raleqbidv 3336 | . . 3 ⊢ (𝑟 = 𝑅 → (∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅)) |
14 | 4, 13 | anbi12d 631 | . 2 ⊢ (𝑟 = 𝑅 → (((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅) ↔ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))) |
15 | df-ufd 31662 | . 2 ⊢ UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)} | |
16 | 14, 15 | elrab2 3627 | 1 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∩ cin 3886 ∅c0 4256 ‘cfv 6433 CRingccrg 19784 RPrimecrpm 19954 AbsValcabv 20076 PrmIdealcprmidl 31610 UFDcufd 31661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ufd 31662 |
This theorem is referenced by: (None) |
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