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Theorem isufd 33555
Description: The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
isufd.i 𝐼 = (PrmIdeal‘𝑅)
isufd.3 𝑃 = (RPrime‘𝑅)
isufd.0 0 = (0g𝑅)
Assertion
Ref Expression
isufd (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖𝑃) ≠ ∅))
Distinct variable group:   𝑅,𝑖
Allowed substitution hints:   𝑃(𝑖)   𝐼(𝑖)   0 (𝑖)

Proof of Theorem isufd
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6904 . . . . 5 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
2 isufd.i . . . . 5 𝐼 = (PrmIdeal‘𝑅)
31, 2eqtr4di 2794 . . . 4 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼)
4 fveq2 6904 . . . . . . 7 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5 isufd.0 . . . . . . 7 0 = (0g𝑅)
64, 5eqtr4di 2794 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = 0 )
76sneqd 4636 . . . . 5 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
87sneqd 4636 . . . 4 (𝑟 = 𝑅 → {{(0g𝑟)}} = {{ 0 }})
93, 8difeq12d 4126 . . 3 (𝑟 = 𝑅 → ((PrmIdeal‘𝑟) ∖ {{(0g𝑟)}}) = (𝐼 ∖ {{ 0 }}))
10 fveq2 6904 . . . . . 6 (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅))
11 isufd.3 . . . . . 6 𝑃 = (RPrime‘𝑅)
1210, 11eqtr4di 2794 . . . . 5 (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃)
1312ineq2d 4219 . . . 4 (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖𝑃))
1413neeq1d 2999 . . 3 (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖𝑃) ≠ ∅))
159, 14raleqbidv 3345 . 2 (𝑟 = 𝑅 → (∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖𝑃) ≠ ∅))
16 df-ufd 33554 . 2 UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
1715, 16elrab2 3694 1 (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖𝑃) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2939  wral 3060  cdif 3947  cin 3949  c0 4332  {csn 4624  cfv 6559  0gc0g 17480  RPrimecrpm 20424  IDomncidom 20685  PrmIdealcprmidl 33450  UFDcufd 33553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-ufd 33554
This theorem is referenced by:  ufdprmidl  33556  ufdidom  33557  pidufd  33558  1arithufdlem4  33562  dfufd2  33565
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