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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 43089 | . . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)) | 
| 3 | 2 | simprbi 496 | . 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen) | 
| 4 | oveq2 7439 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈)) | |
| 5 | lnmlssfg.r | . . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅) | 
| 7 | 6 | eleq1d 2826 | . . 3 ⊢ (𝑎 = 𝑈 → ((𝑀 ↾s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen)) | 
| 8 | 7 | rspcv 3618 | . 2 ⊢ (𝑈 ∈ 𝑆 → (∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen)) | 
| 9 | 3, 8 | mpan9 506 | 1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 ↾s cress 17274 LModclmod 20858 LSubSpclss 20929 LFinGenclfig 43079 LNoeMclnm 43087 | 
| 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-ext 2708 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lnm 43088 | 
| This theorem is referenced by: lnmlsslnm 43093 lnmfg 43094 lnmepi 43097 lmhmlnmsplit 43099 lnrfgtr 43132 | 
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