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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidln1 | Structured version Visualization version GIF version |
Description: One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) |
Ref | Expression |
---|---|
maxidln1.1 | ⊢ 𝐻 = (2nd ‘𝑅) |
maxidln1.2 | ⊢ 𝑈 = (GId‘𝐻) |
Ref | Expression |
---|---|
maxidln1 | ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2735 | . . 3 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
3 | 1, 2 | maxidlnr 38029 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1st ‘𝑅)) |
4 | maxidlidl 38028 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅)) | |
5 | maxidln1.1 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | maxidln1.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
7 | 1, 5, 2, 6 | 1idl 38013 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝑀 ↔ 𝑀 = ran (1st ‘𝑅))) |
8 | 7 | necon3bbid 2976 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1st ‘𝑅))) |
9 | 4, 8 | syldan 591 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (¬ 𝑈 ∈ 𝑀 ↔ 𝑀 ≠ ran (1st ‘𝑅))) |
10 | 3, 9 | mpbird 257 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ran crn 5690 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 GIdcgi 30519 RingOpscrngo 37881 Idlcidl 37994 MaxIdlcmaxidl 37996 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-riota 7388 df-ov 7434 df-1st 8013 df-2nd 8014 df-grpo 30522 df-gid 30523 df-ablo 30574 df-ass 37830 df-exid 37832 df-mgmOLD 37836 df-sgrOLD 37848 df-mndo 37854 df-rngo 37882 df-idl 37997 df-maxidl 37999 |
This theorem is referenced by: maxidln0 38032 |
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