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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnmlssfg | Structured version Visualization version GIF version |
Description: A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lnmlssfg.s | ⊢ 𝑆 = (LSubSp‘𝑀) |
lnmlssfg.r | ⊢ 𝑅 = (𝑀 ↾s 𝑈) |
Ref | Expression |
---|---|
lnmlssfg | ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnmlssfg.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑀) | |
2 | 1 | islnm 40021 | . . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)) |
3 | 2 | simprbi 500 | . 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen) |
4 | oveq2 7143 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈)) | |
5 | lnmlssfg.r | . . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈) | |
6 | 4, 5 | eqtr4di 2851 | . . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅) |
7 | 6 | eleq1d 2874 | . . 3 ⊢ (𝑎 = 𝑈 → ((𝑀 ↾s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen)) |
8 | 7 | rspcv 3566 | . 2 ⊢ (𝑈 ∈ 𝑆 → (∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen)) |
9 | 3, 8 | mpan9 510 | 1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 ↾s cress 16476 LModclmod 19627 LSubSpclss 19696 LFinGenclfig 40011 LNoeMclnm 40019 |
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 |
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-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-lnm 40020 |
This theorem is referenced by: lnmlsslnm 40025 lnmfg 40026 lnmepi 40029 lmhmlnmsplit 40031 lnrfgtr 40064 |
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