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Theorem ismxidl 32623
Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
Hypothesis
Ref Expression
mxidlval.1 𝐡 = (Baseβ€˜π‘…)
Assertion
Ref Expression
ismxidl (𝑅 ∈ Ring β†’ (𝑀 ∈ (MaxIdealβ€˜π‘…) ↔ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))))
Distinct variable groups:   𝑅,𝑗   𝑗,𝑀
Allowed substitution hint:   𝐡(𝑗)
Proof of Theorem ismxidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 mxidlval.1 . . . 4 𝐡 = (Baseβ€˜π‘…)
21mxidlval 32622 . . 3 (𝑅 ∈ Ring β†’ (MaxIdealβ€˜π‘…) = {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))})
32eleq2d 2819 . 2 (𝑅 ∈ Ring β†’ (𝑀 ∈ (MaxIdealβ€˜π‘…) ↔ 𝑀 ∈ {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))}))
4 neeq1 3003 . . . . 5 (𝑖 = 𝑀 β†’ (𝑖 β‰  𝐡 ↔ 𝑀 β‰  𝐡))
5 sseq1 4007 . . . . . . 7 (𝑖 = 𝑀 β†’ (𝑖 βŠ† 𝑗 ↔ 𝑀 βŠ† 𝑗))
6 eqeq2 2744 . . . . . . . 8 (𝑖 = 𝑀 β†’ (𝑗 = 𝑖 ↔ 𝑗 = 𝑀))
76orbi1d 915 . . . . . . 7 (𝑖 = 𝑀 β†’ ((𝑗 = 𝑖 ∨ 𝑗 = 𝐡) ↔ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))
85, 7imbi12d 344 . . . . . 6 (𝑖 = 𝑀 β†’ ((𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)) ↔ (𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡))))
98ralbidv 3177 . . . . 5 (𝑖 = 𝑀 β†’ (βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)) ↔ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡))))
104, 9anbi12d 631 . . . 4 (𝑖 = 𝑀 β†’ ((𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡))) ↔ (𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))))
1110elrab 3683 . . 3 (𝑀 ∈ {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))} ↔ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ (𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))))
12 3anass 1095 . . 3 ((𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡))) ↔ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ (𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))))
1311, 12bitr4i 277 . 2 (𝑀 ∈ {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))} ↔ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡))))
143, 13bitrdi 286 1 (𝑅 ∈ Ring β†’ (𝑀 ∈ (MaxIdealβ€˜π‘…) ↔ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432   βŠ† wss 3948  β€˜cfv 6543  Basecbs 17146  Ringcrg 20058  LIdealclidl 20789  MaxIdealcmxidl 32620
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-mxidl 32621
This theorem is referenced by:  mxidlidl  32624  mxidlnr  32625  mxidlmax  32626  crngmxidl  32630  mxidlirred  32633  ssmxidl  32635  drng0mxidl  32637  opprmxidlabs  32646  qsdrng  32656  zarclssn  32922
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