maxidln0 - Mathbox for Jeff Madsen

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Theorem maxidln0 36908

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 = (1^st ‘𝑅)
maxidln0.2	⊢ 𝐻 = (2^nd ‘𝑅)
maxidln0.3	⊢ 𝑍 = (GId‘𝐺)
maxidln0.4	⊢ 𝑈 = (GId‘𝐻)

**Assertion**
Ref	Expression
maxidln0	⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)

**Proof of Theorem maxidln0**
Step	Hyp	Ref	Expression
1		maxidlidl 36904	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
2		maxidln0.1	. . . . . 6 ⊢ 𝐺 = (1^st ‘𝑅)
3		maxidln0.3	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
4	2, 3	idl0cl 36881	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑀)
5	1, 4	syldan 591	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ∈ 𝑀)
6		maxidln0.2	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
7		maxidln0.4	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
8	6, 7	maxidln1 36907	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)
9		nelneq 2857	. . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈)
10	5, 8, 9	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
11	10	neqned 2947	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈)
12	11	necomd 2996	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ‘cfv 6543 1^st c1st 7972 2^nd c2nd 7973 GIdcgi 29738 RingOpscrngo 36757 Idlcidl 36870 MaxIdlcmaxidl 36872

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-riota 7364 df-ov 7411 df-1st 7974 df-2nd 7975 df-grpo 29741 df-gid 29742 df-ablo 29793 df-ass 36706 df-exid 36708 df-mgmOLD 36712 df-sgrOLD 36724 df-mndo 36730 df-rngo 36758 df-idl 36873 df-maxidl 36875

This theorem is referenced by: (None)

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