Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismxidl Structured version   Visualization version   GIF version
Theorem ismxidl 32578
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 32577 . . 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 2820 . 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 3004 . . . . 5 (𝑖 = 𝑀 β†’ (𝑖 β‰  𝐡 ↔ 𝑀 β‰  𝐡))
5 sseq1 4008 . . . . . . 7 (𝑖 = 𝑀 β†’ (𝑖 βŠ† 𝑗 ↔ 𝑀 βŠ† 𝑗))
6 eqeq2 2745 . . . . . . . 8 (𝑖 = 𝑀 β†’ (𝑗 = 𝑖 ↔ 𝑗 = 𝑀))
76orbi1d 916 . . . . . . 7 (𝑖 = 𝑀 β†’ ((𝑗 = 𝑖 ∨ 𝑗 = 𝐡) ↔ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))
85, 7imbi12d 345 . . . . . 6 (𝑖 = 𝑀 β†’ ((𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)) ↔ (𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡))))
98ralbidv 3178 . . . . 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 632 . . . 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 3684 . . 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 1096 . . 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 278 . 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 287 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433   βŠ† wss 3949  β€˜cfv 6544  Basecbs 17144  Ringcrg 20056  LIdealclidl 20783  MaxIdealcmxidl 32575
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 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-mxidl 32576
This theorem is referenced by:  mxidlidl  32579  mxidlnr  32580  mxidlmax  32581  crngmxidl  32585  mxidlirred  32588  ssmxidl  32590  drng0mxidl  32592  opprmxidlabs  32601  qsdrng  32611  zarclssn  32853
  Copyright terms: Public domain W3C validator