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Theorem prmidlval 31030
 Description: The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
Hypotheses
Ref Expression
prmidlval.1 𝐵 = (Base‘𝑅)
prmidlval.2 · = (.r𝑅)
Assertion
Ref Expression
prmidlval (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
Distinct variable group:   𝑅,𝑎,𝑏,𝑖,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑖,𝑎,𝑏)   · (𝑥,𝑦,𝑖,𝑎,𝑏)

Proof of Theorem prmidlval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 df-prmidl 31029 . 2 PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
2 fveq2 6646 . . 3 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 fveq2 6646 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 prmidlval.1 . . . . . 6 𝐵 = (Base‘𝑅)
53, 4eqtr4di 2851 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
65neeq2d 3047 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ (Base‘𝑟) ↔ 𝑖𝐵))
7 fveq2 6646 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
8 prmidlval.2 . . . . . . . . . . 11 · = (.r𝑅)
97, 8eqtr4di 2851 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = · )
109oveqd 7153 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
1110eleq1d 2874 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑥(.r𝑟)𝑦) ∈ 𝑖 ↔ (𝑥 · 𝑦) ∈ 𝑖))
12112ralbidv 3164 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖))
1312imbi1d 345 . . . . . 6 (𝑟 = 𝑅 → ((∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ (∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
142, 13raleqbidv 3354 . . . . 5 (𝑟 = 𝑅 → (∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ ∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
152, 14raleqbidv 3354 . . . 4 (𝑟 = 𝑅 → (∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
166, 15anbi12d 633 . . 3 (𝑟 = 𝑅 → ((𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))) ↔ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))))
172, 16rabeqbidv 3433 . 2 (𝑟 = 𝑅 → {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
18 id 22 . 2 (𝑅 ∈ Ring → 𝑅 ∈ Ring)
19 eqid 2798 . . 3 {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))}
20 fvexd 6661 . . 3 (𝑅 ∈ Ring → (LIdeal‘𝑅) ∈ V)
2119, 20rabexd 5201 . 2 (𝑅 ∈ Ring → {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} ∈ V)
221, 17, 18, 21fvmptd3 6769 1 (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  .rcmulr 16561  Ringcrg 19294  LIdealclidl 19939  PrmIdealcprmidl 31028 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-ov 7139  df-prmidl 31029 This theorem is referenced by:  isprmidl  31031
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