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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnmlmod | Structured version Visualization version GIF version |
Description: A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
Ref | Expression |
---|---|
lnmlmod | ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
2 | 1 | islnm 39555 | . 2 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀 ↾s 𝑎) ∈ LFinGen)) |
3 | 2 | simplbi 498 | 1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 ↾s cress 16472 LModclmod 19563 LSubSpclss 19632 LFinGenclfig 39545 LNoeMclnm 39553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-lnm 39554 |
This theorem is referenced by: lnmlsslnm 39559 lnmfg 39560 pwslnmlem1 39570 pwslnm 39572 |
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