Theorem isufd 33073

Description: The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.)

Hypotheses

Ref	Expression
isufd.a	⊢ 𝐴 = (AbsVal‘𝑅)
isufd.i	⊢ 𝐼 = (PrmIdeal‘𝑅)
isufd.3	⊢ 𝑃 = (RPrime‘𝑅)

Assertion

Ref	Expression
isufd	⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅)))

Distinct variable group:   𝑅,𝑖

Allowed substitution hints:   𝐴(𝑖)   𝑃(𝑖)   𝐼(𝑖)

Proof of Theorem isufd

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6891	. . . . 5 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
2		isufd.a	. . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅)
3	1, 2	eqtr4di 2789	. . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
4	3	neeq1d 2999	. . 3 ⊢ (𝑟 = 𝑅 → ((AbsVal‘𝑟) ≠ ∅ ↔ 𝐴 ≠ ∅))
5		fveq2 6891	. . . . 5 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
6		isufd.i	. . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅)
7	5, 6	eqtr4di 2789	. . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼)
8		fveq2 6891	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅))
9		isufd.3	. . . . . . 7 ⊢ 𝑃 = (RPrime‘𝑅)
10	8, 9	eqtr4di 2789	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃)
11	10	ineq2d 4212	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖 ∩ 𝑃))
12	11	neeq1d 2999	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅))
13	7, 12	raleqbidv 3341	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))
14	4, 13	anbi12d 630	. 2 ⊢ (𝑟 = 𝑅 → (((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅) ↔ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅)))
15		df-ufd 33072	. 2 ⊢ UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)}
16	14, 15	elrab2 3686	1 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060   ∩ cin 3947  ∅c0 4322  ‘cfv 6543  CRingccrg 20135  RPrimecrpm 20330  AbsValcabv 20655  PrmIdealcprmidl 32994  UFDcufd 33071

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ufd 33072

This theorem is referenced by: (None)