![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
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 35479 | . . . . 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 35456 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝑀) |
5 | 1, 4 | syldan 594 | . . . 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 35482 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) |
9 | nelneq 2914 | . . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈) | |
10 | 5, 8, 9 | syl2anc 587 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈) |
11 | 10 | neqned 2994 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈) |
12 | 11 | necomd 3042 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 1st c1st 7669 2nd 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: (None) |
Copyright terms: Public domain | W3C validator |