Theorem ufdidom 33510

Description: A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypothesis

Ref	Expression
ufdidom.2	⊢ (𝜑 → 𝑅 ∈ UFD)

Assertion

Ref	Expression
ufdidom	⊢ (𝜑 → 𝑅 ∈ IDomn)

Proof of Theorem ufdidom

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ufdidom.2	. 2 ⊢ (𝜑 → 𝑅 ∈ UFD)
2		eqid 2734	. . . 4 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
3		eqid 2734	. . . 4 ⊢ (RPrime‘𝑅) = (RPrime‘𝑅)
4		eqid 2734	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	2, 3, 4	isufd 33508	. . 3 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{(0g‘𝑅)}})(𝑖 ∩ (RPrime‘𝑅)) ≠ ∅))
6	5	simplbi 497	. 2 ⊢ (𝑅 ∈ UFD → 𝑅 ∈ IDomn)
7	1, 6	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ IDomn)

Syntax hints:   → wi 4   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050   ∖ cdif 3928   ∩ cin 3930  ∅c0 4313  {csn 4606  ‘cfv 6541  0_gc0g 17456  RPrimecrpm 20401  IDomncidom 20662  PrmIdealcprmidl 33403  UFDcufd 33506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549  df-ufd 33507

This theorem is referenced by:  1arithufdlem1  33512  1arithufdlem2  33513  1arithufdlem3  33514  1arithufdlem4  33515  dfufd2  33518