Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fglmod Structured version   Visualization version   GIF version

Theorem fglmod 43085
Description: Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Assertion
Ref Expression
fglmod (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Proof of Theorem fglmod
StepHypRef Expression
1 df-lfig 43080 . . 3 LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
21ssrab3 4082 . 2 LFinGen ⊆ LMod
32sseli 3979 1 (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cin 3950  𝒫 cpw 4600  cima 5688  cfv 6561  Fincfn 8985  Basecbs 17247  LModclmod 20858  LSpanclspn 20969  LFinGenclfig 43079
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-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-ss 3968  df-lfig 43080
This theorem is referenced by:  lnrfg  43131
  Copyright terms: Public domain W3C validator