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Theorem isprmidl 33466
Description: The predicate "is a prime ideal". (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
isprmidl (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
Distinct variable groups:   𝑅,𝑎,𝑏,𝑥,𝑦   𝑃,𝑎,𝑏,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑎,𝑏)   · (𝑥,𝑦,𝑎,𝑏)

Proof of Theorem isprmidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 prmidlval.1 . . . . 5 𝐵 = (Base‘𝑅)
2 prmidlval.2 . . . . 5 · = (.r𝑅)
31, 2prmidlval 33465 . . . 4 (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
43eleq2d 2827 . . 3 (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ 𝑃 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))}))
5 neeq1 3003 . . . . 5 (𝑖 = 𝑃 → (𝑖𝐵𝑃𝐵))
6 eleq2 2830 . . . . . . . 8 (𝑖 = 𝑃 → ((𝑥 · 𝑦) ∈ 𝑖 ↔ (𝑥 · 𝑦) ∈ 𝑃))
762ralbidv 3221 . . . . . . 7 (𝑖 = 𝑃 → (∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 ↔ ∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃))
8 sseq2 4010 . . . . . . . 8 (𝑖 = 𝑃 → (𝑎𝑖𝑎𝑃))
9 sseq2 4010 . . . . . . . 8 (𝑖 = 𝑃 → (𝑏𝑖𝑏𝑃))
108, 9orbi12d 919 . . . . . . 7 (𝑖 = 𝑃 → ((𝑎𝑖𝑏𝑖) ↔ (𝑎𝑃𝑏𝑃)))
117, 10imbi12d 344 . . . . . 6 (𝑖 = 𝑃 → ((∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ (∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
12112ralbidv 3221 . . . . 5 (𝑖 = 𝑃 → (∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
135, 12anbi12d 632 . . . 4 (𝑖 = 𝑃 → ((𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))) ↔ (𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
1413elrab 3692 . . 3 (𝑃 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
154, 14bitrdi 287 . 2 (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))))
16 3anass 1095 . 2 ((𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ (𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
1715, 16bitr4di 289 1 (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  Ringcrg 20230  LIdealclidl 21216  PrmIdealcprmidl 33463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-prmidl 33464
This theorem is referenced by:  prmidlnr  33467  prmidl  33468  prmidl2  33469  prmidlidl  33472
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