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Theorem maxidln1 38051
Description: One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
maxidln1.1 𝐻 = (2nd𝑅)
maxidln1.2 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
maxidln1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)

Proof of Theorem maxidln1
StepHypRef Expression
1 eqid 2737 . . 3 (1st𝑅) = (1st𝑅)
2 eqid 2737 . . 3 ran (1st𝑅) = ran (1st𝑅)
31, 2maxidlnr 38049 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1st𝑅))
4 maxidlidl 38048 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
5 maxidln1.1 . . . . 5 𝐻 = (2nd𝑅)
6 maxidln1.2 . . . . 5 𝑈 = (GId‘𝐻)
71, 5, 2, 61idl 38033 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈𝑀𝑀 = ran (1st𝑅)))
87necon3bbid 2978 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈𝑀𝑀 ≠ ran (1st𝑅)))
94, 8syldan 591 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (¬ 𝑈𝑀𝑀 ≠ ran (1st𝑅)))
103, 9mpbird 257 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  ran crn 5686  cfv 6561  1st c1st 8012  2nd c2nd 8013  GIdcgi 30509  RingOpscrngo 37901  Idlcidl 38014  MaxIdlcmaxidl 38016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-ov 7434  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-idl 38017  df-maxidl 38019
This theorem is referenced by:  maxidln0  38052
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