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Theorem mxidlval 33489
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 6906 . . 3 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2 fveq2 6906 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 mxidlval.1 . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54neeq2d 3001 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ (Base‘𝑟) ↔ 𝑖𝐵))
64eqeq2d 2748 . . . . . . 7 (𝑟 = 𝑅 → (𝑗 = (Base‘𝑟) ↔ 𝑗 = 𝐵))
76orbi2d 916 . . . . . 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 3346 . . . 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 3455 . 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 33488 . 2 MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})
13 fvex 6919 . . 3 (LIdeal‘𝑅) ∈ V
1413rabex 5339 . 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 848   = 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:  ismxidl  33490
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