maxidln1 - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidln1		Structured version Visualization version GIF version

Theorem maxidln1 36912

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 2733	. . 3 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2733	. . 3 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	maxidlnr 36910	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1^st ‘𝑅))
4		maxidlidl 36909	. . 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 36894	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝑀 ↔ 𝑀 = ran (1^st ‘𝑅)))
8	7	necon3bbid 2979	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
9	4, 8	syldan 592	. 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ran crn 5678 ‘cfv 6544 1^st c1st 7973 2^nd c2nd 7974 GIdcgi 29743 RingOpscrngo 36762 Idlcidl 36875 MaxIdlcmaxidl 36877

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-riota 7365 df-ov 7412 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gid 29747 df-ablo 29798 df-ass 36711 df-exid 36713 df-mgmOLD 36717 df-sgrOLD 36729 df-mndo 36735 df-rngo 36763 df-idl 36878 df-maxidl 36880

This theorem is referenced by: maxidln0 36913

W3C validator