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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mxidlmaxv | Structured version Visualization version GIF version |
Description: An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
mxidlmaxv.1 | ⊢ 𝐵 = (Base‘𝑅) |
mxidlmaxv.2 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mxidlmaxv.3 | ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) |
mxidlmaxv.4 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
mxidlmaxv.5 | ⊢ (𝜑 → 𝑀 ⊆ 𝐼) |
mxidlmaxv.6 | ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀)) |
Ref | Expression |
---|---|
mxidlmaxv | ⊢ (𝜑 → 𝐼 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mxidlmaxv.2 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | mxidlmaxv.3 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) | |
3 | mxidlmaxv.4 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
4 | mxidlmaxv.5 | . . 3 ⊢ (𝜑 → 𝑀 ⊆ 𝐼) | |
5 | mxidlmaxv.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 5 | mxidlmax 33473 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)) |
7 | 1, 2, 3, 4, 6 | syl22anc 839 | . 2 ⊢ (𝜑 → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)) |
8 | mxidlmaxv.6 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀)) | |
9 | 8 | eldifad 3975 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
10 | 8 | eldifbd 3976 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀) |
11 | nelne1 3037 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝑀) → 𝐼 ≠ 𝑀) | |
12 | 9, 10, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝑀) |
13 | 12 | neneqd 2943 | . 2 ⊢ (𝜑 → ¬ 𝐼 = 𝑀) |
14 | 7, 13 | orcnd 878 | 1 ⊢ (𝜑 → 𝐼 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ⊆ wss 3963 ‘cfv 6563 Basecbs 17245 Ringcrg 20251 LIdealclidl 21234 MaxIdealcmxidl 33467 |
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-pr 5438 |
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-rab 3434 df-v 3480 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-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-iota 6516 df-fun 6565 df-fv 6571 df-mxidl 33468 |
This theorem is referenced by: mxidlirredi 33479 qsdrngilem 33502 |
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