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Theorem mxidlval 31197
Description: The set of maximal ideals of a ring. (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
mxidlval (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})
Distinct variable group:   𝑅,𝑖,𝑗
Allowed substitution hints:   𝐵(𝑖,𝑗)

Proof of Theorem mxidlval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6668 . . 3 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2 fveq2 6668 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 mxidlval.1 . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2791 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54neeq2d 2994 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ (Base‘𝑟) ↔ 𝑖𝐵))
64eqeq2d 2749 . . . . . . 7 (𝑟 = 𝑅 → (𝑗 = (Base‘𝑟) ↔ 𝑗 = 𝐵))
76orbi2d 915 . . . . . 6 (𝑟 = 𝑅 → ((𝑗 = 𝑖𝑗 = (Base‘𝑟)) ↔ (𝑗 = 𝑖𝑗 = 𝐵)))
87imbi2d 344 . . . . 5 (𝑟 = 𝑅 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))) ↔ (𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵))))
91, 8raleqbidv 3303 . . . 4 (𝑟 = 𝑅 → (∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))) ↔ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵))))
105, 9anbi12d 634 . . 3 (𝑟 = 𝑅 → ((𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟)))) ↔ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))))
111, 10rabeqbidv 3386 . 2 (𝑟 = 𝑅 → {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))} = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})
12 df-mxidl 31196 . 2 MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})
13 fvex 6681 . . 3 (LIdeal‘𝑅) ∈ V
1413rabex 5197 . 2 {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))} ∈ V
1511, 12, 14fvmpt 6769 1 (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846   = wceq 1542  wcel 2113  wne 2934  wral 3053  {crab 3057  wss 3841  cfv 6333  Basecbs 16579  Ringcrg 19409  LIdealclidl 20054  MaxIdealcmxidl 31195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-mxidl 31196
This theorem is referenced by:  ismxidl  31198
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