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Theorem ismaxidl 33009
 Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
ismaxidl.1 𝐺 = (1st𝑅)
ismaxidl.2 𝑋 = ran 𝐺
Assertion
Ref Expression
ismaxidl (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
Distinct variable groups:   𝑅,𝑗   𝑗,𝑀
Allowed substitution hints:   𝐺(𝑗)   𝑋(𝑗)

Proof of Theorem ismaxidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 ismaxidl.1 . . . 4 𝐺 = (1st𝑅)
2 ismaxidl.2 . . . 4 𝑋 = ran 𝐺
31, 2maxidlval 33008 . . 3 (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})
43eleq2d 2673 . 2 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))}))
5 neeq1 2844 . . . . 5 (𝑖 = 𝑀 → (𝑖𝑋𝑀𝑋))
6 sseq1 3589 . . . . . . 7 (𝑖 = 𝑀 → (𝑖𝑗𝑀𝑗))
7 eqeq2 2621 . . . . . . . 8 (𝑖 = 𝑀 → (𝑗 = 𝑖𝑗 = 𝑀))
87orbi1d 735 . . . . . . 7 (𝑖 = 𝑀 → ((𝑗 = 𝑖𝑗 = 𝑋) ↔ (𝑗 = 𝑀𝑗 = 𝑋)))
96, 8imbi12d 333 . . . . . 6 (𝑖 = 𝑀 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)) ↔ (𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
109ralbidv 2969 . . . . 5 (𝑖 = 𝑀 → (∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
115, 10anbi12d 743 . . . 4 (𝑖 = 𝑀 → ((𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))) ↔ (𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
1211elrab 3331 . . 3 (𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))} ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
13 3anass 1035 . . 3 ((𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
1412, 13bitr4i 266 . 2 (𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))} ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
154, 14syl6bb 275 1 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900   ⊆ wss 3540  ran crn 5039  ‘cfv 5804  1st c1st 7057  RingOpscrngo 32863  Idlcidl 32976  MaxIdlcmaxidl 32978 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-maxidl 32981 This theorem is referenced by:  maxidlidl  33010  maxidlnr  33011  maxidlmax  33012
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