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Theorem ufdidom 33557
Description: A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025.)
Hypothesis
Ref Expression
ufdidom.2 (𝜑𝑅 ∈ UFD)
Assertion
Ref Expression
ufdidom (𝜑𝑅 ∈ IDomn)

Proof of Theorem ufdidom
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 ufdidom.2 . 2 (𝜑𝑅 ∈ UFD)
2 eqid 2736 . . . 4 (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
3 eqid 2736 . . . 4 (RPrime‘𝑅) = (RPrime‘𝑅)
4 eqid 2736 . . . 4 (0g𝑅) = (0g𝑅)
52, 3, 4isufd 33555 . . 3 (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{(0g𝑅)}})(𝑖 ∩ (RPrime‘𝑅)) ≠ ∅))
65simplbi 497 . 2 (𝑅 ∈ UFD → 𝑅 ∈ IDomn)
71, 6syl 17 1 (𝜑𝑅 ∈ IDomn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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:  1arithufdlem1  33559  1arithufdlem2  33560  1arithufdlem3  33561  1arithufdlem4  33562  dfufd2  33565
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