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Theorem maxidln0 32813
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 32809 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
2 maxidln0.1 . . . . . 6 𝐺 = (1st𝑅)
3 maxidln0.3 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3idl0cl 32786 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍𝑀)
51, 4syldan 485 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑀)
6 maxidln0.2 . . . . 5 𝐻 = (2nd𝑅)
7 maxidln0.4 . . . . 5 𝑈 = (GId‘𝐻)
86, 7maxidln1 32812 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)
9 nelneq 2707 . . . 4 ((𝑍𝑀 ∧ ¬ 𝑈𝑀) → ¬ 𝑍 = 𝑈)
105, 8, 9syl2anc 690 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
1110neqned 2784 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑈)
1211necomd 2832 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  wne 2775  cfv 5786  1st c1st 7030  2nd c2nd 7031  GIdcgi 26490  RingOpscrngo 32662  Idlcidl 32775  MaxIdlcmaxidl 32777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-fo 5792  df-fv 5794  df-riota 6485  df-ov 6526  df-1st 7032  df-2nd 7033  df-grpo 26493  df-gid 26494  df-ablo 26548  df-ass 32611  df-exid 32613  df-mgmOLD 32617  df-sgrOLD 32629  df-mndo 32635  df-rngo 32663  df-idl 32778  df-maxidl 32780
This theorem is referenced by: (None)
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