Theorem ufdidom 33699

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 2761	. . . 4 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
3		eqid 2761	. . . 4 ⊢ (RPrime‘𝑅) = (RPrime‘𝑅)
4		eqid 2761	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	2, 3, 4	isufd 33697	. . 3 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{(0g‘𝑅)}})(𝑖 ∩ (RPrime‘𝑅)) ≠ ∅))
6	5	simplbi 500	. 2 ⊢ (𝑅 ∈ UFD → 𝑅 ∈ IDomn)
7	1, 6	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ IDomn)

Syntax hints:   → wi 4   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075   ∖ cdif 3901   ∩ cin 3903  ∅c0 4285  {csn 4581  ‘cfv 6517  0_gc0g 17451  RPrimecrpm 20460  IDomncidom 20722  PrmIdealcprmidl 33582  UFDcufd 33695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-ufd 33696

This theorem is referenced by:  1arithufdlem1  33701  1arithufdlem2  33702  1arithufdlem3  33703  1arithufdlem4  33704  dfufd2  33707