maxidln1 - Mathbox for Jeff Madsen

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Theorem maxidln1 38073

Description: One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)

**Hypotheses**
Ref	Expression
maxidln1.1	⊢ 𝐻 = (2^nd ‘𝑅)
maxidln1.2	⊢ 𝑈 = (GId‘𝐻)

**Assertion**
Ref	Expression
maxidln1	⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)

**Proof of Theorem maxidln1**
Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2736	. . 3 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	maxidlnr 38071	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1^st ‘𝑅))
4		maxidlidl 38070	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
5		maxidln1.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
6		maxidln1.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
7	1, 5, 2, 6	1idl 38055	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝑀 ↔ 𝑀 = ran (1^st ‘𝑅)))
8	7	necon3bbid 2970	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
9	4, 8	syldan 591	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
10	3, 9	mpbird 257	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ran crn 5660 ‘cfv 6536 1^st c1st 7991 2^nd c2nd 7992 GIdcgi 30476 RingOpscrngo 37923 Idlcidl 38036 MaxIdlcmaxidl 38038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-riota 7367 df-ov 7413 df-1st 7993 df-2nd 7994 df-grpo 30479 df-gid 30480 df-ablo 30531 df-ass 37872 df-exid 37874 df-mgmOLD 37878 df-sgrOLD 37890 df-mndo 37896 df-rngo 37924 df-idl 38039 df-maxidl 38041

This theorem is referenced by: maxidln0 38074

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