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 6668 | . . . . 5 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
2 | isufd.a | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
3 | 1, 2 | eqtr4di 2791 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) |
4 | 3 | neeq1d 2993 | . . 3 ⊢ (𝑟 = 𝑅 → ((AbsVal‘𝑟) ≠ ∅ ↔ 𝐴 ≠ ∅)) |
5 | fveq2 6668 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅)) | |
6 | isufd.i | . . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅) | |
7 | 5, 6 | eqtr4di 2791 | . . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼) |
8 | fveq2 6668 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅)) | |
9 | isufd.3 | . . . . . . 7 ⊢ 𝑃 = (RPrime‘𝑅) | |
10 | 8, 9 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃) |
11 | 10 | ineq2d 4101 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖 ∩ 𝑃)) |
12 | 11 | neeq1d 2993 | . . . 4 ⊢ (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅)) |
13 | 7, 12 | raleqbidv 3303 | . . 3 ⊢ (𝑟 = 𝑅 → (∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅)) |
14 | 4, 13 | anbi12d 634 | . 2 ⊢ (𝑟 = 𝑅 → (((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅) ↔ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))) |
15 | df-ufd 31226 | . 2 ⊢ UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)} | |
16 | 14, 15 | elrab2 3588 | 1 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∀wral 3053 ∩ cin 3840 ∅c0 4209 ‘cfv 6333 CRingccrg 19410 RPrimecrpm 19577 AbsValcabv 19699 PrmIdealcprmidl 31174 UFDcufd 31225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rab 3062 df-v 3399 df-un 3846 df-in 3848 df-ss 3858 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-iota 6291 df-fv 6341 df-ufd 31226 |
This theorem is referenced by: (None) |
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