Theorem fglmod 36661

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

Step	Hyp	Ref	Expression
1		df-lfig 36656	. . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2		ssrab2 3650	. . 3 ⊢ {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} ⊆ LMod
3	1, 2	eqsstri 3598	. 2 ⊢ LFinGen ⊆ LMod
4	3	sseli 3564	1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 1977  {crab 2900   ∩ cin 3539  𝒫 cpw 4108   “ cima 5041  ‘cfv 5804  Fincfn 7841  Basecbs 15695  LModclmod 18686  LSpanclspn 18792  LFinGenclfig 36655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554  df-lfig 36656

This theorem is referenced by:  lnrfg  36708