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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 2731 | . . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2731 | . . 3 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
3 | 1, 2 | maxidlnr 36715 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ ran (1st ‘𝑅)) |
4 | maxidlidl 36714 | . . 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 36699 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝑀 ↔ 𝑀 = ran (1st ‘𝑅))) |
8 | 7 | necon3bbid 2977 | . . 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 256 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ran crn 5670 ‘cfv 6532 1st c1st 7955 2nd c2nd 7956 GIdcgi 29606 RingOpscrngo 36567 Idlcidl 36680 MaxIdlcmaxidl 36682 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fo 6538 df-fv 6540 df-riota 7349 df-ov 7396 df-1st 7957 df-2nd 7958 df-grpo 29609 df-gid 29610 df-ablo 29661 df-ass 36516 df-exid 36518 df-mgmOLD 36522 df-sgrOLD 36534 df-mndo 36540 df-rngo 36568 df-idl 36683 df-maxidl 36685 |
This theorem is referenced by: maxidln0 36718 |
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