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Theorem prmidlval 32257
Description: The class of prime ideals of a ring 𝑅. (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
prmidlval (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)))})
Distinct variable group:   𝑅,π‘Ž,𝑏,𝑖,π‘₯,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   Β· (π‘₯,𝑦,𝑖,π‘Ž,𝑏)

Proof of Theorem prmidlval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 df-prmidl 32256 . 2 PrmIdeal = (π‘Ÿ ∈ Ring ↦ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘Ÿ)βˆ€π‘ ∈ (LIdealβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
2 fveq2 6843 . . 3 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
3 fveq2 6843 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
4 prmidlval.1 . . . . . 6 𝐡 = (Baseβ€˜π‘…)
53, 4eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
65neeq2d 3001 . . . 4 (π‘Ÿ = 𝑅 β†’ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ↔ 𝑖 β‰  𝐡))
7 fveq2 6843 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
8 prmidlval.2 . . . . . . . . . . 11 Β· = (.rβ€˜π‘…)
97, 8eqtr4di 2791 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
109oveqd 7375 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (π‘₯(.rβ€˜π‘Ÿ)𝑦) = (π‘₯ Β· 𝑦))
1110eleq1d 2819 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 ↔ (π‘₯ Β· 𝑦) ∈ 𝑖))
12112ralbidv 3209 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 ↔ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖))
1312imbi1d 342 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ((βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))))
142, 13raleqbidv 3318 . . . . 5 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘ ∈ (LIdealβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))))
152, 14raleqbidv 3318 . . . 4 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘Ž ∈ (LIdealβ€˜π‘Ÿ)βˆ€π‘ ∈ (LIdealβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)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) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))))
166, 15anbi12d 632 . . 3 (π‘Ÿ = 𝑅 β†’ ((𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘Ÿ)βˆ€π‘ ∈ (LIdealβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))) ↔ (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)))))
172, 16rabeqbidv 3423 . 2 (π‘Ÿ = 𝑅 β†’ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘Ÿ)βˆ€π‘ ∈ (LIdealβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))} = {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)))})
18 id 22 . 2 (𝑅 ∈ Ring β†’ 𝑅 ∈ Ring)
19 eqid 2733 . . 3 {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)))} = {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)))}
20 fvexd 6858 . . 3 (𝑅 ∈ Ring β†’ (LIdealβ€˜π‘…) ∈ V)
2119, 20rabexd 5291 . 2 (𝑅 ∈ Ring β†’ {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)))} ∈ V)
221, 17, 18, 21fvmptd3 6972 1 (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)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  {crab 3406  Vcvv 3444   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:  isprmidl  32258
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