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Theorem prmidl 30967
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 3390 . . . . 5 (𝑏 = 𝐽 → (∀𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃))
21ralbidv 3185 . . . 4 (𝑏 = 𝐽 → (∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃))
3 sseq1 3968 . . . . 5 (𝑏 = 𝐽 → (𝑏𝑃𝐽𝑃))
43orbi2d 913 . . . 4 (𝑏 = 𝐽 → ((𝐼𝑃𝑏𝑃) ↔ (𝐼𝑃𝐽𝑃)))
52, 4imbi12d 348 . . 3 (𝑏 = 𝐽 → ((∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝑏𝑃)) ↔ (∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝐽𝑃))))
6 raleq 3390 . . . . . 6 (𝑎 = 𝐼 → (∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 ↔ ∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃))
7 sseq1 3968 . . . . . . 7 (𝑎 = 𝐼 → (𝑎𝑃𝐼𝑃))
87orbi1d 914 . . . . . 6 (𝑎 = 𝐼 → ((𝑎𝑃𝑏𝑃) ↔ (𝐼𝑃𝑏𝑃)))
96, 8imbi12d 348 . . . . 5 (𝑎 = 𝐼 → ((∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ (∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝑏𝑃))))
109ralbidv 3185 . . . 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 30965 . . . . . . 7 (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
1413biimpa 480 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
1514simp3d 1141 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
1615adantr 484 . . . 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 770 . . . 4 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐼 ∈ (LIdeal‘𝑅))
1810, 16, 17rspcdva 3602 . . 3 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → ∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝐼𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝑏𝑃)))
19 simprr 772 . . 3 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → 𝐽 ∈ (LIdeal‘𝑅))
205, 18, 19rspcdva 3602 . 2 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) → (∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃 → (𝐼𝑃𝐽𝑃)))
2120imp 410 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 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wral 3126  wss 3910  cfv 6328  (class class class)co 7130  Basecbs 16462  .rcmulr 16545  Ringcrg 19276  LIdealclidl 19918  PrmIdealcprmidl 30962
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-prmidl 30963
This theorem is referenced by:  idlmulssprm  30969  isprmidlc  30974
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