maxidln0 - Mathbox for Jeff Madsen

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Theorem maxidln0 38024

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 38020	. . . . 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 37997	. . . . 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 38023	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)
9		nelneq 2852	. . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈)
10	5, 8, 9	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
11	10	neqned 2932	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈)
12	11	necomd 2980	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ‘cfv 6486 1^st c1st 7929 2^nd c2nd 7930 GIdcgi 30452 RingOpscrngo 37873 Idlcidl 37986 MaxIdlcmaxidl 37988

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 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fo 6492 df-fv 6494 df-riota 7310 df-ov 7356 df-1st 7931 df-2nd 7932 df-grpo 30455 df-gid 30456 df-ablo 30507 df-ass 37822 df-exid 37824 df-mgmOLD 37828 df-sgrOLD 37840 df-mndo 37846 df-rngo 37874 df-idl 37989 df-maxidl 37991

This theorem is referenced by: (None)

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