Theorem ufdidom 33602

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 2736	. . . 4 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
3		eqid 2736	. . . 4 ⊢ (RPrime‘𝑅) = (RPrime‘𝑅)
4		eqid 2736	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	2, 3, 4	isufd 33600	. . 3 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{(0g‘𝑅)}})(𝑖 ∩ (RPrime‘𝑅)) ≠ ∅))
6	5	simplbi 496	. 2 ⊢ (𝑅 ∈ UFD → 𝑅 ∈ IDomn)
7	1, 6	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ IDomn)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051   ∖ cdif 3886   ∩ cin 3888  ∅c0 4273  {csn 4567  ‘cfv 6498  0_gc0g 17402  RPrimecrpm 20412  IDomncidom 20670  PrmIdealcprmidl 33495  UFDcufd 33598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ufd 33599

This theorem is referenced by:  1arithufdlem1  33604  1arithufdlem2  33605  1arithufdlem3  33606  1arithufdlem4  33607  dfufd2  33610