Theorem ufdidom 33507

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 2731	. . . 4 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
3		eqid 2731	. . . 4 ⊢ (RPrime‘𝑅) = (RPrime‘𝑅)
4		eqid 2731	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	2, 3, 4	isufd 33505	. . 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 2111   ≠ wne 2928  ∀wral 3047   ∖ cdif 3894   ∩ cin 3896  ∅c0 4280  {csn 4573  ‘cfv 6481  0_gc0g 17343  RPrimecrpm 20350  IDomncidom 20608  PrmIdealcprmidl 33400  UFDcufd 33503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ufd 33504

This theorem is referenced by:  1arithufdlem1  33509  1arithufdlem2  33510  1arithufdlem3  33511  1arithufdlem4  33512  dfufd2  33515