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Theorem maxidln1 34378
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 2825 . . 3 (1st𝑅) = (1st𝑅)
2 eqid 2825 . . 3 ran (1st𝑅) = ran (1st𝑅)
31, 2maxidlnr 34376 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1st𝑅))
4 maxidlidl 34375 . . 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 34360 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈𝑀𝑀 = ran (1st𝑅)))
87necon3bbid 3036 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈𝑀𝑀 ≠ ran (1st𝑅)))
94, 8syldan 585 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (¬ 𝑈𝑀𝑀 ≠ ran (1st𝑅)))
103, 9mpbird 249 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wne 2999  ran crn 5343  cfv 6123  1st c1st 7426  2nd c2nd 7427  GIdcgi 27889  RingOpscrngo 34228  Idlcidl 34341  MaxIdlcmaxidl 34343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fo 6129  df-fv 6131  df-riota 6866  df-ov 6908  df-1st 7428  df-2nd 7429  df-grpo 27892  df-gid 27893  df-ablo 27944  df-ass 34177  df-exid 34179  df-mgmOLD 34183  df-sgrOLD 34195  df-mndo 34201  df-rngo 34229  df-idl 34344  df-maxidl 34346
This theorem is referenced by:  maxidln0  34379
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