| Mathbox for Stefan O'Rear |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lnmfg | Structured version Visualization version GIF version | ||
| Description: A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
| Ref | Expression |
|---|---|
| lnmfg | ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 2 | 1 | ressid 17290 | . 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾s (Base‘𝑀)) = 𝑀) |
| 3 | lnmlmod 43091 | . . . 4 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
| 5 | 1, 4 | lss1 20936 | . . . 4 ⊢ (𝑀 ∈ LMod → (Base‘𝑀) ∈ (LSubSp‘𝑀)) |
| 6 | 3, 5 | syl 17 | . . 3 ⊢ (𝑀 ∈ LNoeM → (Base‘𝑀) ∈ (LSubSp‘𝑀)) |
| 7 | eqid 2737 | . . . 4 ⊢ (𝑀 ↾s (Base‘𝑀)) = (𝑀 ↾s (Base‘𝑀)) | |
| 8 | 4, 7 | lnmlssfg 43092 | . . 3 ⊢ ((𝑀 ∈ LNoeM ∧ (Base‘𝑀) ∈ (LSubSp‘𝑀)) → (𝑀 ↾s (Base‘𝑀)) ∈ LFinGen) |
| 9 | 6, 8 | mpdan 687 | . 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾s (Base‘𝑀)) ∈ LFinGen) |
| 10 | 2, 9 | eqeltrrd 2842 | 1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-ress 17275 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 df-lss 20930 df-lnm 43088 |
| This theorem is referenced by: (None) |
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