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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 2779 | . . 3 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
2 | 1 | islnm 39070 | . 2 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀 ↾s 𝑎) ∈ LFinGen)) |
3 | 2 | simplbi 490 | 1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 ∀wral 3089 ‘cfv 6188 (class class class)co 6976 ↾s cress 16340 LModclmod 19356 LSubSpclss 19425 LFinGenclfig 39060 LNoeMclnm 39068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-iota 6152 df-fv 6196 df-ov 6979 df-lnm 39069 |
This theorem is referenced by: lnmlsslnm 39074 lnmfg 39075 pwslnmlem1 39085 pwslnm 39087 |
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