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Theorem maxidln0 35441
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 35437 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
2 maxidln0.1 . . . . . 6 𝐺 = (1st𝑅)
3 maxidln0.3 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3idl0cl 35414 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍𝑀)
51, 4syldan 594 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑀)
6 maxidln0.2 . . . . 5 𝐻 = (2nd𝑅)
7 maxidln0.4 . . . . 5 𝑈 = (GId‘𝐻)
86, 7maxidln1 35440 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)
9 nelneq 2938 . . . 4 ((𝑍𝑀 ∧ ¬ 𝑈𝑀) → ¬ 𝑍 = 𝑈)
105, 8, 9syl2anc 587 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
1110neqned 3018 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑈)
1211necomd 3066 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  wne 3011  cfv 6334  1st c1st 7673  2nd c2nd 7674  GIdcgi 28271  RingOpscrngo 35290  Idlcidl 35403  MaxIdlcmaxidl 35405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fo 6340  df-fv 6342  df-riota 7098  df-ov 7143  df-1st 7675  df-2nd 7676  df-grpo 28274  df-gid 28275  df-ablo 28326  df-ass 35239  df-exid 35241  df-mgmOLD 35245  df-sgrOLD 35257  df-mndo 35263  df-rngo 35291  df-idl 35406  df-maxidl 35408
This theorem is referenced by: (None)
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