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Theorem maxidlval 36907
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 2791 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5938 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 maxidlval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2791 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87neeq2d 3002 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ ran (1st𝑟) ↔ 𝑖𝑋))
97eqeq2d 2744 . . . . . . 7 (𝑟 = 𝑅 → (𝑗 = ran (1st𝑟) ↔ 𝑗 = 𝑋))
109orbi2d 915 . . . . . 6 (𝑟 = 𝑅 → ((𝑗 = 𝑖𝑗 = ran (1st𝑟)) ↔ (𝑗 = 𝑖𝑗 = 𝑋)))
1110imbi2d 341 . . . . 5 (𝑟 = 𝑅 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))) ↔ (𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))))
121, 11raleqbidv 3343 . . . 4 (𝑟 = 𝑅 → (∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))))
138, 12anbi12d 632 . . 3 (𝑟 = 𝑅 → ((𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟)))) ↔ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))))
141, 13rabeqbidv 3450 . 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 36880 . 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 5333 . 2 {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))} ∈ V
1814, 15, 17fvmpt 6999 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 397  wo 846   = wceq 1542  wcel 2107  wne 2941  wral 3062  {crab 3433  wss 3949  ran crn 5678  cfv 6544  1st c1st 7973  RingOpscrngo 36762  Idlcidl 36875  MaxIdlcmaxidl 36877
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 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-maxidl 36880
This theorem is referenced by:  ismaxidl  36908
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