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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 31042 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))) |
3 | 2 | biimpa 480 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))) |
4 | 3 | simp2d 1140 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ⊆ wss 3881 ‘cfv 6324 Basecbs 16475 Ringcrg 19290 LIdealclidl 19935 MaxIdealcmxidl 31039 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-mxidl 31040 |
This theorem is referenced by: mxidln1 31046 mxidlprm 31048 |
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