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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 6651 | . . . . 5 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
2 | isufd.a | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
3 | 1, 2 | eqtr4di 2812 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) |
4 | 3 | neeq1d 3008 | . . 3 ⊢ (𝑟 = 𝑅 → ((AbsVal‘𝑟) ≠ ∅ ↔ 𝐴 ≠ ∅)) |
5 | fveq2 6651 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅)) | |
6 | isufd.i | . . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅) | |
7 | 5, 6 | eqtr4di 2812 | . . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼) |
8 | fveq2 6651 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅)) | |
9 | isufd.3 | . . . . . . 7 ⊢ 𝑃 = (RPrime‘𝑅) | |
10 | 8, 9 | eqtr4di 2812 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃) |
11 | 10 | ineq2d 4113 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖 ∩ 𝑃)) |
12 | 11 | neeq1d 3008 | . . . 4 ⊢ (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅)) |
13 | 7, 12 | raleqbidv 3317 | . . 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 31168 | . 2 ⊢ UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)} | |
16 | 14, 15 | elrab2 3603 | 1 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ≠ wne 2949 ∀wral 3068 ∩ cin 3853 ∅c0 4221 ‘cfv 6328 CRingccrg 19351 RPrimecrpm 19518 AbsValcabv 19640 PrmIdealcprmidl 31116 UFDcufd 31167 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rab 3077 df-v 3409 df-un 3859 df-in 3861 df-ss 3871 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-iota 6287 df-fv 6336 df-ufd 31168 |
This theorem is referenced by: (None) |
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