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Theorem maxidlidl 37754
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 2726 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2726 . . . 4 ran (1st𝑅) = ran (1st𝑅)
31, 2ismaxidl 37753 . . 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, 4bitrdi 286 . 2 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀 ≠ ran (1st𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = ran (1st𝑅)))))))
65simprbda 497 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wss 3948  ran crn 5675  cfv 6545  1st c1st 7992  RingOpscrngo 37607  Idlcidl 37720  MaxIdlcmaxidl 37722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3466  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6497  df-fun 6547  df-fv 6553  df-maxidl 37725
This theorem is referenced by:  maxidln1  37757  maxidln0  37758
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