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Theorem isufd 32315
Description: The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
isufd.a 𝐴 = (AbsValβ€˜π‘…)
isufd.i 𝐼 = (PrmIdealβ€˜π‘…)
isufd.3 𝑃 = (RPrimeβ€˜π‘…)
Assertion
Ref Expression
isufd (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 β‰  βˆ… ∧ βˆ€π‘– ∈ 𝐼 (𝑖 ∩ 𝑃) β‰  βˆ…)))
Distinct variable group:   𝑅,𝑖
Allowed substitution hints:   𝐴(𝑖)   𝑃(𝑖)   𝐼(𝑖)

Proof of Theorem isufd
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6846 . . . . 5 (π‘Ÿ = 𝑅 β†’ (AbsValβ€˜π‘Ÿ) = (AbsValβ€˜π‘…))
2 isufd.a . . . . 5 𝐴 = (AbsValβ€˜π‘…)
31, 2eqtr4di 2791 . . . 4 (π‘Ÿ = 𝑅 β†’ (AbsValβ€˜π‘Ÿ) = 𝐴)
43neeq1d 3000 . . 3 (π‘Ÿ = 𝑅 β†’ ((AbsValβ€˜π‘Ÿ) β‰  βˆ… ↔ 𝐴 β‰  βˆ…))
5 fveq2 6846 . . . . 5 (π‘Ÿ = 𝑅 β†’ (PrmIdealβ€˜π‘Ÿ) = (PrmIdealβ€˜π‘…))
6 isufd.i . . . . 5 𝐼 = (PrmIdealβ€˜π‘…)
75, 6eqtr4di 2791 . . . 4 (π‘Ÿ = 𝑅 β†’ (PrmIdealβ€˜π‘Ÿ) = 𝐼)
8 fveq2 6846 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (RPrimeβ€˜π‘Ÿ) = (RPrimeβ€˜π‘…))
9 isufd.3 . . . . . . 7 𝑃 = (RPrimeβ€˜π‘…)
108, 9eqtr4di 2791 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (RPrimeβ€˜π‘Ÿ) = 𝑃)
1110ineq2d 4176 . . . . 5 (π‘Ÿ = 𝑅 β†’ (𝑖 ∩ (RPrimeβ€˜π‘Ÿ)) = (𝑖 ∩ 𝑃))
1211neeq1d 3000 . . . 4 (π‘Ÿ = 𝑅 β†’ ((𝑖 ∩ (RPrimeβ€˜π‘Ÿ)) β‰  βˆ… ↔ (𝑖 ∩ 𝑃) β‰  βˆ…))
137, 12raleqbidv 3318 . . 3 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘– ∈ (PrmIdealβ€˜π‘Ÿ)(𝑖 ∩ (RPrimeβ€˜π‘Ÿ)) β‰  βˆ… ↔ βˆ€π‘– ∈ 𝐼 (𝑖 ∩ 𝑃) β‰  βˆ…))
144, 13anbi12d 632 . 2 (π‘Ÿ = 𝑅 β†’ (((AbsValβ€˜π‘Ÿ) β‰  βˆ… ∧ βˆ€π‘– ∈ (PrmIdealβ€˜π‘Ÿ)(𝑖 ∩ (RPrimeβ€˜π‘Ÿ)) β‰  βˆ…) ↔ (𝐴 β‰  βˆ… ∧ βˆ€π‘– ∈ 𝐼 (𝑖 ∩ 𝑃) β‰  βˆ…)))
15 df-ufd 32314 . 2 UFD = {π‘Ÿ ∈ CRing ∣ ((AbsValβ€˜π‘Ÿ) β‰  βˆ… ∧ βˆ€π‘– ∈ (PrmIdealβ€˜π‘Ÿ)(𝑖 ∩ (RPrimeβ€˜π‘Ÿ)) β‰  βˆ…)}
1614, 15elrab2 3652 1 (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 β‰  βˆ… ∧ βˆ€π‘– ∈ 𝐼 (𝑖 ∩ 𝑃) β‰  βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061   ∩ cin 3913  βˆ…c0 4286  β€˜cfv 6500  CRingccrg 19973  RPrimecrpm 20151  AbsValcabv 20318  PrmIdealcprmidl 32262  UFDcufd 32313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ufd 32314
This theorem is referenced by: (None)
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