maxidln1 - Mathbox for Jeff Madsen

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Theorem maxidln1 35482

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 2798	. . 3 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2798	. . 3 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	maxidlnr 35480	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1^st ‘𝑅))
4		maxidlidl 35479	. . 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 35464	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝑀 ↔ 𝑀 = ran (1^st ‘𝑅)))
8	7	necon3bbid 3024	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
9	4, 8	syldan 594	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1^st ‘𝑅)))
10	3, 9	mpbird 260	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ran crn 5520 ‘cfv 6324 1^st c1st 7669 2^nd c2nd 7670 GIdcgi 28273 RingOpscrngo 35332 Idlcidl 35445 MaxIdlcmaxidl 35447

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-riota 7093 df-ov 7138 df-1st 7671 df-2nd 7672 df-grpo 28276 df-gid 28277 df-ablo 28328 df-ass 35281 df-exid 35283 df-mgmOLD 35287 df-sgrOLD 35299 df-mndo 35305 df-rngo 35333 df-idl 35448 df-maxidl 35450

This theorem is referenced by: maxidln0 35483

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