maxidln0 - Mathbox for Jeff Madsen

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Theorem maxidln0 38084

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 38080	. . . . 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 38057	. . . . 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 38083	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)
9		nelneq 2855	. . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈)
10	5, 8, 9	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
11	10	neqned 2935	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈)
12	11	necomd 2983	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ‘cfv 6481 1^st c1st 7919 2^nd c2nd 7920 GIdcgi 30465 RingOpscrngo 37933 Idlcidl 38046 MaxIdlcmaxidl 38048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-riota 7303 df-ov 7349 df-1st 7921 df-2nd 7922 df-grpo 30468 df-gid 30469 df-ablo 30520 df-ass 37882 df-exid 37884 df-mgmOLD 37888 df-sgrOLD 37900 df-mndo 37906 df-rngo 37934 df-idl 38049 df-maxidl 38051

This theorem is referenced by: (None)

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