Theorem isufd 33894

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

Hypotheses

Ref	Expression
isufd.i	⊢ 𝐼 = (PrmIdeal‘𝑅)
isufd.3	⊢ 𝑃 = (RPrime‘𝑅)
isufd.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
isufd	⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖 ∩ 𝑃) ≠ ∅))

Distinct variable group:   𝑅,𝑖

Allowed substitution hints:   𝑃(𝑖)   𝐼(𝑖)   0 (𝑖)

Proof of Theorem isufd

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . 5 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
2		isufd.i	. . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅)
3	1, 2	eqtr4di 2818	. . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼)
4		fveq2 6885	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
5		isufd.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
6	4, 5	eqtr4di 2818	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
7	6	sneqd 4603	. . . . 5 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
8	7	sneqd 4603	. . . 4 ⊢ (𝑟 = 𝑅 → {{(0g‘𝑟)}} = {{ 0 }})
9	3, 8	difeq12d 4082	. . 3 ⊢ (𝑟 = 𝑅 → ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}}) = (𝐼 ∖ {{ 0 }}))
10		fveq2 6885	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅))
11		isufd.3	. . . . . 6 ⊢ 𝑃 = (RPrime‘𝑅)
12	10, 11	eqtr4di 2818	. . . . 5 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃)
13	12	ineq2d 4173	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖 ∩ 𝑃))
14	13	neeq1d 3019	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅))
15	9, 14	raleqbidv 3340	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖 ∩ 𝑃) ≠ ∅))
16		df-ufd 33893	. 2 ⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
17	15, 16	elrab2 3656	1 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖 ∩ 𝑃) ≠ ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081   ∖ cdif 3903   ∩ cin 3905  ∅c0 4286  {csn 4591  ‘cfv 6540  0_gc0g 17514  RPrimecrpm 20560  IDomncidom 20842  PrmIdealcprmidl 21510  UFDcufd 33892

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ufd 33893

This theorem is used by:  ufdprmidl  33895  ufdidom  33896  pidufd  33897  1arithufdlem4  33901  dfufd2  33904