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Theorem mxidlval 33469
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 6907 . . 3 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2 fveq2 6907 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 mxidlval.1 . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54neeq2d 2999 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ (Base‘𝑟) ↔ 𝑖𝐵))
64eqeq2d 2746 . . . . . . 7 (𝑟 = 𝑅 → (𝑗 = (Base‘𝑟) ↔ 𝑗 = 𝐵))
76orbi2d 915 . . . . . 6 (𝑟 = 𝑅 → ((𝑗 = 𝑖𝑗 = (Base‘𝑟)) ↔ (𝑗 = 𝑖𝑗 = 𝐵)))
87imbi2d 340 . . . . 5 (𝑟 = 𝑅 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))) ↔ (𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵))))
91, 8raleqbidv 3344 . . . 4 (𝑟 = 𝑅 → (∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))) ↔ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵))))
105, 9anbi12d 632 . . 3 (𝑟 = 𝑅 → ((𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟)))) ↔ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))))
111, 10rabeqbidv 3452 . 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 33468 . 2 MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})
13 fvex 6920 . . 3 (LIdeal‘𝑅) ∈ V
1413rabex 5345 . 2 {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))} ∈ V
1511, 12, 14fvmpt 7016 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 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wral 3059  {crab 3433  wss 3963  cfv 6563  Basecbs 17245  Ringcrg 20251  LIdealclidl 21234  MaxIdealcmxidl 33467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-mxidl 33468
This theorem is referenced by:  ismxidl  33470
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