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Theorem maxidlidl 35437
Description: A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
Assertion
Ref Expression
maxidlidl ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))

Proof of Theorem maxidlidl
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2822 . . . 4 ran (1st𝑅) = ran (1st𝑅)
31, 2ismaxidl 35436 . . 3 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ ran (1st𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = ran (1st𝑅))))))
4 3anass 1092 . . 3 ((𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ ran (1st𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = ran (1st𝑅)))) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀 ≠ ran (1st𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = ran (1st𝑅))))))
53, 4syl6bb 290 . 2 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀 ≠ ran (1st𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = ran (1st𝑅)))))))
65simprbda 502 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2114  wne 3011  wral 3130  wss 3908  ran crn 5533  cfv 6334  1st c1st 7673  RingOpscrngo 35290  Idlcidl 35403  MaxIdlcmaxidl 35405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fv 6342  df-maxidl 35408
This theorem is referenced by:  maxidln1  35440  maxidln0  35441
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