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Theorem maxidlval 37569
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 6892 . . 3 (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2 fveq2 6892 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 maxidlval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3eqtr4di 2783 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5934 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 maxidlval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2783 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87neeq2d 2991 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ ran (1st𝑟) ↔ 𝑖𝑋))
97eqeq2d 2736 . . . . . . 7 (𝑟 = 𝑅 → (𝑗 = ran (1st𝑟) ↔ 𝑗 = 𝑋))
109orbi2d 913 . . . . . 6 (𝑟 = 𝑅 → ((𝑗 = 𝑖𝑗 = ran (1st𝑟)) ↔ (𝑗 = 𝑖𝑗 = 𝑋)))
1110imbi2d 339 . . . . 5 (𝑟 = 𝑅 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))) ↔ (𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))))
121, 11raleqbidv 3330 . . . 4 (𝑟 = 𝑅 → (∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))))
138, 12anbi12d 630 . . 3 (𝑟 = 𝑅 → ((𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟)))) ↔ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))))
141, 13rabeqbidv 3437 . 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 37542 . 2 MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))))})
16 fvex 6905 . . 3 (Idl‘𝑅) ∈ V
1716rabex 5329 . 2 {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))} ∈ V
1814, 15, 17fvmpt 7000 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 394  wo 845   = wceq 1533  wcel 2098  wne 2930  wral 3051  {crab 3419  wss 3939  ran crn 5673  cfv 6543  1st c1st 7989  RingOpscrngo 37424  Idlcidl 37537  MaxIdlcmaxidl 37539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-maxidl 37542
This theorem is referenced by:  ismaxidl  37570
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