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Mathbox for Jeff Madsen |
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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 34464 | . . . . 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 34441 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑀) |
5 | 1, 4 | syldan 585 | . . . 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 34467 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) |
9 | nelneq 2883 | . . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈) | |
10 | 5, 8, 9 | syl2anc 579 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈) |
11 | 10 | neqned 2976 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈) |
12 | 11 | necomd 3024 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ‘cfv 6135 1st c1st 7443 2nd c2nd 7444 GIdcgi 27917 RingOpscrngo 34317 Idlcidl 34430 MaxIdlcmaxidl 34432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fo 6141 df-fv 6143 df-riota 6883 df-ov 6925 df-1st 7445 df-2nd 7446 df-grpo 27920 df-gid 27921 df-ablo 27972 df-ass 34266 df-exid 34268 df-mgmOLD 34272 df-sgrOLD 34284 df-mndo 34290 df-rngo 34318 df-idl 34433 df-maxidl 34435 |
This theorem is referenced by: (None) |
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