Theorem isufd 33511

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 6901	. . . . 5 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
2		isufd.i	. . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅)
3	1, 2	eqtr4di 2791	. . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = 𝐼)
4		fveq2 6901	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
5		isufd.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
6	4, 5	eqtr4di 2791	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
7	6	sneqd 4642	. . . . 5 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
8	7	sneqd 4642	. . . 4 ⊢ (𝑟 = 𝑅 → {{(0g‘𝑟)}} = {{ 0 }})
9	3, 8	difeq12d 4137	. . 3 ⊢ (𝑟 = 𝑅 → ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}}) = (𝐼 ∖ {{ 0 }}))
10		fveq2 6901	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = (RPrime‘𝑅))
11		isufd.3	. . . . . 6 ⊢ 𝑃 = (RPrime‘𝑅)
12	10, 11	eqtr4di 2791	. . . . 5 ⊢ (𝑟 = 𝑅 → (RPrime‘𝑟) = 𝑃)
13	12	ineq2d 4228	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ∩ (RPrime‘𝑟)) = (𝑖 ∩ 𝑃))
14	13	neeq1d 2996	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅))
15	9, 14	raleqbidv 3342	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅ ↔ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖 ∩ 𝑃) ≠ ∅))
16		df-ufd 33510	. 2 ⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
17	15, 16	elrab2 3698	1 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖 ∩ 𝑃) ≠ ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1535   ∈ wcel 2104   ≠ wne 2936  ∀wral 3057   ∖ cdif 3960   ∩ cin 3962  ∅c0 4339  {csn 4630  ‘cfv 6558  0_gc0g 17475  RPrimecrpm 20434  IDomncidom 20691  PrmIdealcprmidl 33406  UFDcufd 33509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ne 2937  df-ral 3058  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-iota 6510  df-fv 6566  df-ufd 33510

This theorem is referenced by:  ufdprmidl  33512  ufdidom  33513  pidufd  33514  1arithufdlem4  33518  dfufd2  33521