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Theorem isprmidl 32258
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 𝐡 = (Baseβ€˜π‘…)
prmidlval.2 Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
isprmidl (𝑅 ∈ Ring β†’ (𝑃 ∈ (PrmIdealβ€˜π‘…) ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ 𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
Distinct variable groups:   𝑅,π‘Ž,𝑏,π‘₯,𝑦   𝑃,π‘Ž,𝑏,π‘₯,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦,π‘Ž,𝑏)   Β· (π‘₯,𝑦,π‘Ž,𝑏)

Proof of Theorem isprmidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 prmidlval.1 . . . . 5 𝐡 = (Baseβ€˜π‘…)
2 prmidlval.2 . . . . 5 Β· = (.rβ€˜π‘…)
31, 2prmidlval 32257 . . . 4 (𝑅 ∈ Ring β†’ (PrmIdealβ€˜π‘…) = {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
43eleq2d 2820 . . 3 (𝑅 ∈ Ring β†’ (𝑃 ∈ (PrmIdealβ€˜π‘…) ↔ 𝑃 ∈ {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))}))
5 neeq1 3003 . . . . 5 (𝑖 = 𝑃 β†’ (𝑖 β‰  𝐡 ↔ 𝑃 β‰  𝐡))
6 eleq2 2823 . . . . . . . 8 (𝑖 = 𝑃 β†’ ((π‘₯ Β· 𝑦) ∈ 𝑖 ↔ (π‘₯ Β· 𝑦) ∈ 𝑃))
762ralbidv 3209 . . . . . . 7 (𝑖 = 𝑃 β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 ↔ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃))
8 sseq2 3971 . . . . . . . 8 (𝑖 = 𝑃 β†’ (π‘Ž βŠ† 𝑖 ↔ π‘Ž βŠ† 𝑃))
9 sseq2 3971 . . . . . . . 8 (𝑖 = 𝑃 β†’ (𝑏 βŠ† 𝑖 ↔ 𝑏 βŠ† 𝑃))
108, 9orbi12d 918 . . . . . . 7 (𝑖 = 𝑃 β†’ ((π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖) ↔ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
117, 10imbi12d 345 . . . . . 6 (𝑖 = 𝑃 β†’ ((βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
12112ralbidv 3209 . . . . 5 (𝑖 = 𝑃 β†’ (βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
135, 12anbi12d 632 . . . 4 (𝑖 = 𝑃 β†’ ((𝑖 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))) ↔ (𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
1413elrab 3646 . . 3 (𝑃 ∈ {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))} ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ (𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
154, 14bitrdi 287 . 2 (𝑅 ∈ Ring β†’ (𝑃 ∈ (PrmIdealβ€˜π‘…) ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ (𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))))
16 3anass 1096 . 2 ((𝑃 ∈ (LIdealβ€˜π‘…) ∧ 𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))) ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ (𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
1715, 16bitr4di 289 1 (𝑅 ∈ Ring β†’ (𝑃 ∈ (PrmIdealβ€˜π‘…) ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ 𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  {crab 3406   βŠ† wss 3911  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  .rcmulr 17139  Ringcrg 19969  LIdealclidl 20647  PrmIdealcprmidl 32255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-prmidl 32256
This theorem is referenced by:  prmidlnr  32259  prmidl  32260  prmidl2  32261  prmidlidl  32264
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