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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 43066 | . . 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 2793 | . . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅) |
7 | 6 | eleq1d 2824 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 ↾s cress 17274 LModclmod 20875 LSubSpclss 20947 LFinGenclfig 43056 LNoeMclnm 43064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-lnm 43065 |
This theorem is referenced by: lnmlsslnm 43070 lnmfg 43071 lnmepi 43074 lmhmlnmsplit 43076 lnrfgtr 43109 |
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