maxidln0 - Mathbox for Jeff Madsen

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Theorem maxidln0 37426

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 37422	. . . . 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 37399	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑀)
5	1, 4	syldan 590	. . . 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 37425	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)
9		nelneq 2851	. . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈)
10	5, 8, 9	syl2anc 583	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
11	10	neqned 2941	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈)
12	11	necomd 2990	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ‘cfv 6537 1^st c1st 7972 2^nd c2nd 7973 GIdcgi 30252 RingOpscrngo 37275 Idlcidl 37388 MaxIdlcmaxidl 37390

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-riota 7361 df-ov 7408 df-1st 7974 df-2nd 7975 df-grpo 30255 df-gid 30256 df-ablo 30307 df-ass 37224 df-exid 37226 df-mgmOLD 37230 df-sgrOLD 37242 df-mndo 37248 df-rngo 37276 df-idl 37391 df-maxidl 37393

This theorem is referenced by: (None)

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