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Theorem maxidln0 37559
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 37555 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
2 maxidln0.1 . . . . . 6 𝐺 = (1st𝑅)
3 maxidln0.3 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3idl0cl 37532 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍𝑀)
51, 4syldan 589 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑀)
6 maxidln0.2 . . . . 5 𝐻 = (2nd𝑅)
7 maxidln0.4 . . . . 5 𝑈 = (GId‘𝐻)
86, 7maxidln1 37558 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)
9 nelneq 2853 . . . 4 ((𝑍𝑀 ∧ ¬ 𝑈𝑀) → ¬ 𝑍 = 𝑈)
105, 8, 9syl2anc 582 . . 3 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
1110neqned 2944 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍𝑈)
1211necomd 2993 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2937  cfv 6553  1st c1st 7999  2nd c2nd 8000  GIdcgi 30328  RingOpscrngo 37408  Idlcidl 37521  MaxIdlcmaxidl 37523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-riota 7382  df-ov 7429  df-1st 8001  df-2nd 8002  df-grpo 30331  df-gid 30332  df-ablo 30383  df-ass 37357  df-exid 37359  df-mgmOLD 37363  df-sgrOLD 37375  df-mndo 37381  df-rngo 37409  df-idl 37524  df-maxidl 37526
This theorem is referenced by: (None)
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