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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 39697 | . . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)) |
3 | 2 | simprbi 499 | . 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen) |
4 | oveq2 7164 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈)) | |
5 | lnmlssfg.r | . . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈) | |
6 | 4, 5 | syl6eqr 2874 | . . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅) |
7 | 6 | eleq1d 2897 | . . 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 509 | 1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 ↾s cress 16484 LModclmod 19634 LSubSpclss 19703 LFinGenclfig 39687 LNoeMclnm 39695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-lnm 39696 |
This theorem is referenced by: lnmlsslnm 39701 lnmfg 39702 lnmepi 39705 lmhmlnmsplit 39707 lnrfgtr 39740 |
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