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Theorem maxidln1 36506
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 36504 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1st𝑅))
4 maxidlidl 36503 . . 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 36488 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈𝑀𝑀 = ran (1st𝑅)))
87necon3bbid 2982 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈𝑀𝑀 ≠ ran (1st𝑅)))
94, 8syldan 592 . 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 205  wa 397   = wceq 1542  wcel 2107  wne 2944  ran crn 5635  cfv 6497  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356  Idlcidl 36469  MaxIdlcmaxidl 36471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-riota 7314  df-ov 7361  df-1st 7922  df-2nd 7923  df-grpo 29438  df-gid 29439  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-idl 36472  df-maxidl 36474
This theorem is referenced by:  maxidln0  36507
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