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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 35321 | . . . . 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 35298 | . . . . 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 35324 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) |
9 | nelneq 2939 | . . . 4 ⊢ ((𝑍 ∈ 𝑀 ∧ ¬ 𝑈 ∈ 𝑀) → ¬ 𝑍 = 𝑈) | |
10 | 5, 8, 9 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑍 = 𝑈) |
11 | 10 | neqned 3025 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑍 ≠ 𝑈) |
12 | 11 | necomd 3073 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 GIdcgi 28269 RingOpscrngo 35174 Idlcidl 35287 MaxIdlcmaxidl 35289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-riota 7116 df-ov 7161 df-1st 7691 df-2nd 7692 df-grpo 28272 df-gid 28273 df-ablo 28324 df-ass 35123 df-exid 35125 df-mgmOLD 35129 df-sgrOLD 35141 df-mndo 35147 df-rngo 35175 df-idl 35290 df-maxidl 35292 |
This theorem is referenced by: (None) |
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