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Theorem isufd 31227
Description: The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
isufd.a 𝐴 = (AbsVal‘𝑅)
isufd.i 𝐼 = (PrmIdeal‘𝑅)
isufd.3 𝑃 = (RPrime‘𝑅)
Assertion
Ref Expression
isufd (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅)))
Distinct variable group:   𝑅,𝑖
Allowed substitution hints:   𝐴(𝑖)   𝑃(𝑖)   𝐼(𝑖)

Proof of Theorem isufd
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6668 . . . . 5 (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
2 isufd.a . . . . 5 𝐴 = (AbsVal‘𝑅)
31, 2eqtr4di 2791 . . . 4 (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
43neeq1d 2993 . . 3 (𝑟 = 𝑅 → ((AbsVal‘𝑟) ≠ ∅ ↔ 𝐴 ≠ ∅))
5 fveq2 6668 . . . . 5 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
6 isufd.i . . . . 5 𝐼 = (PrmIdeal‘𝑅)
75, 6eqtr4di 2791 . . . 4 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼)
8 fveq2 6668 . . . . . . 7 (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅))
9 isufd.3 . . . . . . 7 𝑃 = (RPrime‘𝑅)
108, 9eqtr4di 2791 . . . . . 6 (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃)
1110ineq2d 4101 . . . . 5 (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖𝑃))
1211neeq1d 2993 . . . 4 (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖𝑃) ≠ ∅))
137, 12raleqbidv 3303 . . 3 (𝑟 = 𝑅 → (∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅))
144, 13anbi12d 634 . 2 (𝑟 = 𝑅 → (((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅) ↔ (𝐴 ≠ ∅ ∧ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅)))
15 df-ufd 31226 . 2 UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)}
1614, 15elrab2 3588 1 (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅)))
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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