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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 37647 | . . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)) |
3 | 2 | simprbi 480 | . 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen) |
4 | oveq2 6658 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈)) | |
5 | lnmlssfg.r | . . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈) | |
6 | 4, 5 | syl6eqr 2674 | . . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅) |
7 | 6 | eleq1d 2686 | . . 3 ⊢ (𝑎 = 𝑈 → ((𝑀 ↾s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen)) |
8 | 7 | rspcv 3305 | . 2 ⊢ (𝑈 ∈ 𝑆 → (∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen)) |
9 | 3, 8 | mpan9 486 | 1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 ↾s cress 15858 LModclmod 18863 LSubSpclss 18932 LFinGenclfig 37637 LNoeMclnm 37645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-lnm 37646 |
This theorem is referenced by: lnmlsslnm 37651 lnmfg 37652 lnmepi 37655 lmhmlnmsplit 37657 lnrfgtr 37690 |
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