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Theorem isufd 31663
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 6774 . . . . 5 (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
2 isufd.a . . . . 5 𝐴 = (AbsVal‘𝑅)
31, 2eqtr4di 2796 . . . 4 (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
43neeq1d 3003 . . 3 (𝑟 = 𝑅 → ((AbsVal‘𝑟) ≠ ∅ ↔ 𝐴 ≠ ∅))
5 fveq2 6774 . . . . 5 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
6 isufd.i . . . . 5 𝐼 = (PrmIdeal‘𝑅)
75, 6eqtr4di 2796 . . . 4 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼)
8 fveq2 6774 . . . . . . 7 (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅))
9 isufd.3 . . . . . . 7 𝑃 = (RPrime‘𝑅)
108, 9eqtr4di 2796 . . . . . 6 (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃)
1110ineq2d 4146 . . . . 5 (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖𝑃))
1211neeq1d 3003 . . . 4 (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖𝑃) ≠ ∅))
137, 12raleqbidv 3336 . . 3 (𝑟 = 𝑅 → (∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅))
144, 13anbi12d 631 . 2 (𝑟 = 𝑅 → (((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅) ↔ (𝐴 ≠ ∅ ∧ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅)))
15 df-ufd 31662 . 2 UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)}
1614, 15elrab2 3627 1 (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅)))
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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