Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  maxidln0 Structured version   Visualization version   GIF version

Theorem maxidln0 34468
 Description: A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
maxidln0.1 𝐺 = (1st𝑅)
maxidln0.2 𝐻 = (2nd𝑅)
maxidln0.3 𝑍 = (GId‘𝐺)
maxidln0.4 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
maxidln0 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈𝑍)

Proof of Theorem maxidln0
StepHypRef Expression
1 maxidlidl 34464 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
2 maxidln0.1 . . . . . 6 𝐺 = (1st𝑅)
3 maxidln0.3 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3idl0cl 34441 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍𝑀)
51, 4syldan 585 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑀)
6 maxidln0.2 . . . . 5 𝐻 = (2nd𝑅)
7 maxidln0.4 . . . . 5 𝑈 = (GId‘𝐻)
86, 7maxidln1 34467 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)
9 nelneq 2883 . . . 4 ((𝑍𝑀 ∧ ¬ 𝑈𝑀) → ¬ 𝑍 = 𝑈)
105, 8, 9syl2anc 579 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
1110neqned 2976 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑈)
1211necomd 3024 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈𝑍)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ‘cfv 6135  1st c1st 7443  2nd c2nd 7444  GIdcgi 27917  RingOpscrngo 34317  Idlcidl 34430  MaxIdlcmaxidl 34432 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143  df-riota 6883  df-ov 6925  df-1st 7445  df-2nd 7446  df-grpo 27920  df-gid 27921  df-ablo 27972  df-ass 34266  df-exid 34268  df-mgmOLD 34272  df-sgrOLD 34284  df-mndo 34290  df-rngo 34318  df-idl 34433  df-maxidl 34435 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator