Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prmidl Structured version   Visualization version   GIF version

Theorem prmidl 33448
Description: The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
Hypotheses
Ref Expression
prmidlval.1 𝐵 = (Base‘𝑅)
prmidlval.2 · = (.r𝑅)
Assertion
Ref Expression
prmidl ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼𝑃𝐽𝑃))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑃,𝑦   𝑥,𝐼   𝑥,𝐽,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   · (𝑥,𝑦)   𝐼(𝑦)

Proof of Theorem prmidl
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3321 . . . . 5 (𝑏 = 𝐽 → (∀𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃))
21ralbidv 3176 . . . 4 (𝑏 = 𝐽 → (∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃))
3 sseq1 4021 . . . . 5 (𝑏 = 𝐽 → (𝑏𝑃𝐽𝑃))
43orbi2d 915 . . . 4 (𝑏 = 𝐽 → ((𝐼𝑃𝑏𝑃) ↔ (𝐼𝑃𝐽𝑃)))
52, 4imbi12d 344 . . 3 (𝑏 = 𝐽 → ((∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝑏𝑃)) ↔ (∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝐽𝑃))))
6 raleq 3321 . . . . . 6 (𝑎 = 𝐼 → (∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃))
7 sseq1 4021 . . . . . . 7 (𝑎 = 𝐼 → (𝑎𝑃𝐼𝑃))
87orbi1d 916 . . . . . 6 (𝑎 = 𝐼 → ((𝑎𝑃𝑏𝑃) ↔ (𝐼𝑃𝑏𝑃)))
96, 8imbi12d 344 . . . . 5 (𝑎 = 𝐼 → ((∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ (∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝑏𝑃))))
109ralbidv 3176 . . . 4 (𝑎 = 𝐼 → (∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ ∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝑏𝑃))))
11 prmidlval.1 . . . . . . . 8 𝐵 = (Base‘𝑅)
12 prmidlval.2 . . . . . . . 8 · = (.r𝑅)
1311, 12isprmidl 33446 . . . . . . 7 (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
1413biimpa 476 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
1514simp3d 1143 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
1615adantr 480 . . . 4 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
17 simprl 771 . . . 4 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐼 ∈ (LIdeal‘𝑅))
1810, 16, 17rspcdva 3623 . . 3 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → ∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝑏𝑃)))
19 simprr 773 . . 3 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐽 ∈ (LIdeal‘𝑅))
205, 18, 19rspcdva 3623 . 2 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → (∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝐽𝑃)))
2120imp 406 1 ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼𝑃𝐽𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  Ringcrg 20251  LIdealclidl 21234  PrmIdealcprmidl 33443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-prmidl 33444
This theorem is referenced by:  idlmulssprm  33450  isprmidlc  33455
  Copyright terms: Public domain W3C validator