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Theorem ismxidl 33490
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 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
ismxidl (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
Distinct variable groups:   𝑅,𝑗   𝑗,𝑀
Allowed substitution hint:   𝐵(𝑗)

Proof of Theorem ismxidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 mxidlval.1 . . . 4 𝐵 = (Base‘𝑅)
21mxidlval 33489 . . 3 (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})
32eleq2d 2827 . 2 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))}))
4 neeq1 3003 . . . . 5 (𝑖 = 𝑀 → (𝑖𝐵𝑀𝐵))
5 sseq1 4009 . . . . . . 7 (𝑖 = 𝑀 → (𝑖𝑗𝑀𝑗))
6 eqeq2 2749 . . . . . . . 8 (𝑖 = 𝑀 → (𝑗 = 𝑖𝑗 = 𝑀))
76orbi1d 917 . . . . . . 7 (𝑖 = 𝑀 → ((𝑗 = 𝑖𝑗 = 𝐵) ↔ (𝑗 = 𝑀𝑗 = 𝐵)))
85, 7imbi12d 344 . . . . . 6 (𝑖 = 𝑀 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)) ↔ (𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))))
98ralbidv 3178 . . . . 5 (𝑖 = 𝑀 → (∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)) ↔ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))))
104, 9anbi12d 632 . . . 4 (𝑖 = 𝑀 → ((𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵))) ↔ (𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
1110elrab 3692 . . 3 (𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))} ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ (𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
12 3anass 1095 . . 3 ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ (𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
1311, 12bitr4i 278 . 2 (𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))} ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))))
143, 13bitrdi 287 1 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  wss 3951  cfv 6561  Basecbs 17247  Ringcrg 20230  LIdealclidl 21216  MaxIdealcmxidl 33487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-mxidl 33488
This theorem is referenced by:  mxidlidl  33491  mxidlnr  33492  mxidlmax  33493  crngmxidl  33497  mxidlirred  33500  ssmxidl  33502  drng0mxidl  33504  opprmxidlabs  33515  qsdrng  33525  zarclssn  33872
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