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Theorem maxidlval 35934
Description: The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
maxidlval.1 𝐺 = (1st𝑅)
maxidlval.2 𝑋 = ran 𝐺
Assertion
Ref Expression
maxidlval (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})
Distinct variable group:   𝑅,𝑖,𝑗
Allowed substitution hints:   𝐺(𝑖,𝑗)   𝑋(𝑖,𝑗)

Proof of Theorem maxidlval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6717 . . 3 (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2 fveq2 6717 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 maxidlval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3eqtr4di 2796 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5807 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 maxidlval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2796 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87neeq2d 3001 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ ran (1st𝑟) ↔ 𝑖𝑋))
97eqeq2d 2748 . . . . . . 7 (𝑟 = 𝑅 → (𝑗 = ran (1st𝑟) ↔ 𝑗 = 𝑋))
109orbi2d 916 . . . . . 6 (𝑟 = 𝑅 → ((𝑗 = 𝑖𝑗 = ran (1st𝑟)) ↔ (𝑗 = 𝑖𝑗 = 𝑋)))
1110imbi2d 344 . . . . 5 (𝑟 = 𝑅 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))) ↔ (𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))))
121, 11raleqbidv 3313 . . . 4 (𝑟 = 𝑅 → (∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))))
138, 12anbi12d 634 . . 3 (𝑟 = 𝑅 → ((𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟)))) ↔ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))))
141, 13rabeqbidv 3396 . 2 (𝑟 = 𝑅 → {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))))} = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})
15 df-maxidl 35907 . 2 MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))))})
16 fvex 6730 . . 3 (Idl‘𝑅) ∈ V
1716rabex 5225 . 2 {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))} ∈ V
1814, 15, 17fvmpt 6818 1 (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 847   = wceq 1543  wcel 2110  wne 2940  wral 3061  {crab 3065  wss 3866  ran crn 5552  cfv 6380  1st c1st 7759  RingOpscrngo 35789  Idlcidl 35902  MaxIdlcmaxidl 35904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fv 6388  df-maxidl 35907
This theorem is referenced by:  ismaxidl  35935
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