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Theorem fglmod 37123
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 37118 . . 3 LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2 ssrab2 3666 . . 3 {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} ⊆ LMod
31, 2eqsstri 3614 . 2 LFinGen ⊆ LMod
43sseli 3579 1 (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  {crab 2911  cin 3554  𝒫 cpw 4130  cima 5077  cfv 5847  Fincfn 7899  Basecbs 15781  LModclmod 18784  LSpanclspn 18890  LFinGenclfig 37117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-in 3562  df-ss 3569  df-lfig 37118
This theorem is referenced by:  lnrfg  37170
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