Theorem ufdidom 33744

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 2765	. . . 4 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
3		eqid 2765	. . . 4 ⊢ (RPrime‘𝑅) = (RPrime‘𝑅)
4		eqid 2765	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	2, 3, 4	isufd 33742	. . 3 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{(0g‘𝑅)}})(𝑖 ∩ (RPrime‘𝑅)) ≠ ∅))
6	5	simplbi 501	. 2 ⊢ (𝑅 ∈ UFD → 𝑅 ∈ IDomn)
7	1, 6	syl 18	1 ⊢ (𝜑 → 𝑅 ∈ IDomn)

Syntax hints:   → wi 4   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079   ∖ cdif 3904   ∩ cin 3906  ∅c0 4288  {csn 4585  ‘cfv 6525  0_gc0g 17480  RPrimecrpm 20502  IDomncidom 20766  PrmIdealcprmidl 21419  UFDcufd 33740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-ufd 33741

This theorem is referenced by:  1arithufdlem1  33746  1arithufdlem2  33747  1arithufdlem3  33748  1arithufdlem4  33749  dfufd2  33752