| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mxidlnr | Structured version Visualization version GIF version | ||
| Description: A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| Ref | Expression |
|---|---|
| mxidlval.1 | ⊢ 𝐵 = (Base‘𝑅) |
| Ref | Expression |
|---|---|
| mxidlnr | ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mxidlval.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | 1 | ismxidl 33477 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))) |
| 3 | 2 | biimpa 476 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))) |
| 4 | 3 | simp2d 1144 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∀wral 3060 ⊆ wss 3950 ‘cfv 6559 Basecbs 17243 Ringcrg 20226 LIdealclidl 21208 MaxIdealcmxidl 33474 |
| 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 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| 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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6512 df-fun 6561 df-fv 6567 df-mxidl 33475 |
| This theorem is referenced by: mxidln1 33481 mxidlprm 33485 mxidlirredi 33486 drngmxidl 33492 opprmxidlabs 33502 qsdrngi 33510 |
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