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Theorem prmidlidl 33451
Description: A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
Assertion
Ref Expression
prmidlidl ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))

Proof of Theorem prmidlidl
Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2734 . . . 4 (.r𝑅) = (.r𝑅)
31, 2isprmidl 33445 . . 3 (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ (Base‘𝑅) ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
43biimpa 476 . 2 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ (Base‘𝑅) ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
54simp1d 1141 1 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086  wcel 2105  wne 2937  wral 3058  wss 3962  cfv 6562  (class class class)co 7430  Basecbs 17244  .rcmulr 17298  Ringcrg 20250  LIdealclidl 21233  PrmIdealcprmidl 33442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-prmidl 33443
This theorem is referenced by:  prmidlssidl  33452  isprmidlc  33454  rhmpreimaprmidl  33458  qsidomlem2  33460  pidufd  33550  dfufd2lem  33556  dfufd2  33557  zarcls0  33828  zarcls1  33829  zarclsiin  33831  zarclssn  33833  zart0  33839
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