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Theorem ismaxidl 34797
 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 34796 . . 3 (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})
43eleq2d 2844 . 2 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))}))
5 neeq1 3022 . . . . 5 (𝑖 = 𝑀 → (𝑖𝑋𝑀𝑋))
6 sseq1 3875 . . . . . . 7 (𝑖 = 𝑀 → (𝑖𝑗𝑀𝑗))
7 eqeq2 2782 . . . . . . . 8 (𝑖 = 𝑀 → (𝑗 = 𝑖𝑗 = 𝑀))
87orbi1d 901 . . . . . . 7 (𝑖 = 𝑀 → ((𝑗 = 𝑖𝑗 = 𝑋) ↔ (𝑗 = 𝑀𝑗 = 𝑋)))
96, 8imbi12d 337 . . . . . 6 (𝑖 = 𝑀 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)) ↔ (𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
109ralbidv 3140 . . . . 5 (𝑖 = 𝑀 → (∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
115, 10anbi12d 622 . . . 4 (𝑖 = 𝑀 → ((𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))) ↔ (𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
1211elrab 3588 . . 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 1077 . . 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 270 . 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 279 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 198   ∧ wa 387   ∨ wo 834   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ≠ wne 2960  ∀wral 3081  {crab 3085   ⊆ wss 3822  ran crn 5404  ‘cfv 6185  1st c1st 7497  RingOpscrngo 34651  Idlcidl 34764  MaxIdlcmaxidl 34766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-iota 6149  df-fun 6187  df-fv 6193  df-maxidl 34769 This theorem is referenced by:  maxidlidl  34798  maxidlnr  34799  maxidlmax  34800
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