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Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidln0 | Structured version Visualization version GIF version |
Description: A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
Ref | Expression |
---|---|
maxidln0.1 | ⊢ 𝐺 = (1st ‘𝑅) |
maxidln0.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
maxidln0.3 | ⊢ 𝑍 = (GId‘𝐺) |
maxidln0.4 | ⊢ 𝑈 = (GId‘𝐻) |
Ref | Expression |
---|---|
maxidln0 | ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | maxidlidl 35313 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅)) | |
2 | maxidln0.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | maxidln0.3 | . . . . . 6 ⊢ 𝑍 = (GId‘𝐺) | |
4 | 2, 3 | idl0cl 35290 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑀) |
5 | 1, 4 | syldan 593 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ∈ 𝑀) |
6 | maxidln0.2 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
7 | maxidln0.4 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
8 | 6, 7 | maxidln1 35316 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) |
9 | nelneq 2937 | . . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈) | |
10 | 5, 8, 9 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈) |
11 | 10 | neqned 3023 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈) |
12 | 11 | necomd 3071 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6350 1st c1st 7681 2nd c2nd 7682 GIdcgi 28261 RingOpscrngo 35166 Idlcidl 35279 MaxIdlcmaxidl 35281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fo 6356 df-fv 6358 df-riota 7108 df-ov 7153 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ablo 28316 df-ass 35115 df-exid 35117 df-mgmOLD 35121 df-sgrOLD 35133 df-mndo 35139 df-rngo 35167 df-idl 35282 df-maxidl 35284 |
This theorem is referenced by: (None) |
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