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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 33493 | . . 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 3963 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
| 10 | 8 | eldifbd 3964 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀) |
| 11 | nelne1 3039 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝑀) → 𝐼 ≠ 𝑀) | |
| 12 | 9, 10, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝑀) |
| 13 | 12 | neneqd 2945 | . 2 ⊢ (𝜑 → ¬ 𝐼 = 𝑀) |
| 14 | 7, 13 | orcnd 879 | 1 ⊢ (𝜑 → 𝐼 = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 ‘cfv 6561 Basecbs 17247 Ringcrg 20230 LIdealclidl 21216 MaxIdealcmxidl 33487 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-mxidl 33488 |
| This theorem is referenced by: mxidlirredi 33499 qsdrngilem 33522 |
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