maxidln0 - Mathbox for Jeff Madsen

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Theorem maxidln0 38380

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 38376	. . . . 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 38353	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑀)
5	1, 4	syldan 592	. . . 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 38379	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)
9		nelneq 2861	. . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈)
10	5, 8, 9	syl2anc 585	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈)
11	10	neqned 2940	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈)
12	11	necomd 2988	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ‘cfv 6492 1^st c1st 7933 2^nd c2nd 7934 GIdcgi 30576 RingOpscrngo 38229 Idlcidl 38342 MaxIdlcmaxidl 38344

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fo 6498 df-fv 6500 df-riota 7317 df-ov 7363 df-1st 7935 df-2nd 7936 df-grpo 30579 df-gid 30580 df-ablo 30631 df-ass 38178 df-exid 38180 df-mgmOLD 38184 df-sgrOLD 38196 df-mndo 38202 df-rngo 38230 df-idl 38345 df-maxidl 38347

This theorem is referenced by: (None)

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