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Theorem mxidlval 32285
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 𝐡 = (Baseβ€˜π‘…)
Assertion
Ref Expression
mxidlval (𝑅 ∈ Ring β†’ (MaxIdealβ€˜π‘…) = {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))})
Distinct variable group:   𝑅,𝑖,𝑗
Allowed substitution hints:   𝐡(𝑖,𝑗)

Proof of Theorem mxidlval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6846 . . 3 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
2 fveq2 6846 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
3 mxidlval.1 . . . . . 6 𝐡 = (Baseβ€˜π‘…)
42, 3eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
54neeq2d 3001 . . . 4 (π‘Ÿ = 𝑅 β†’ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ↔ 𝑖 β‰  𝐡))
64eqeq2d 2744 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝑗 = (Baseβ€˜π‘Ÿ) ↔ 𝑗 = 𝐡))
76orbi2d 915 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ((𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ)) ↔ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))
87imbi2d 341 . . . . 5 (π‘Ÿ = 𝑅 β†’ ((𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))) ↔ (𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡))))
91, 8raleqbidv 3318 . . . 4 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))) ↔ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡))))
105, 9anbi12d 632 . . 3 (π‘Ÿ = 𝑅 β†’ ((𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ)))) ↔ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))))
111, 10rabeqbidv 3423 . 2 (π‘Ÿ = 𝑅 β†’ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))))} = {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))})
12 df-mxidl 32284 . 2 MaxIdeal = (π‘Ÿ ∈ Ring ↦ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))))})
13 fvex 6859 . . 3 (LIdealβ€˜π‘…) ∈ V
1413rabex 5293 . 2 {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))} ∈ V
1511, 12, 14fvmpt 6952 1 (𝑅 ∈ Ring β†’ (MaxIdealβ€˜π‘…) = {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝐡)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  {crab 3406   βŠ† wss 3914  β€˜cfv 6500  Basecbs 17091  Ringcrg 19972  LIdealclidl 20676  MaxIdealcmxidl 32283
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 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-mxidl 32284
This theorem is referenced by:  ismxidl  32286
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