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Theorem ismaxidl 37421
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 37420 . . 3 (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})
43eleq2d 2813 . 2 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))}))
5 neeq1 2997 . . . . 5 (𝑖 = 𝑀 → (𝑖𝑋𝑀𝑋))
6 sseq1 4002 . . . . . . 7 (𝑖 = 𝑀 → (𝑖𝑗𝑀𝑗))
7 eqeq2 2738 . . . . . . . 8 (𝑖 = 𝑀 → (𝑗 = 𝑖𝑗 = 𝑀))
87orbi1d 913 . . . . . . 7 (𝑖 = 𝑀 → ((𝑗 = 𝑖𝑗 = 𝑋) ↔ (𝑗 = 𝑀𝑗 = 𝑋)))
96, 8imbi12d 344 . . . . . 6 (𝑖 = 𝑀 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)) ↔ (𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
109ralbidv 3171 . . . . 5 (𝑖 = 𝑀 → (∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
115, 10anbi12d 630 . . . 4 (𝑖 = 𝑀 → ((𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋))) ↔ (𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
1211elrab 3678 . . 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 1092 . . 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 278 . 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, 14bitrdi 287 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 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wne 2934  wral 3055  {crab 3426  wss 3943  ran crn 5670  cfv 6537  1st c1st 7972  RingOpscrngo 37275  Idlcidl 37388  MaxIdlcmaxidl 37390
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-maxidl 37393
This theorem is referenced by:  maxidlidl  37422  maxidlnr  37423  maxidlmax  37424
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