Theorem fglmod 42367

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 42362	. . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2	1	ssrab3 4073	. 2 ⊢ LFinGen ⊆ LMod
3	2	sseli 3971	1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2098   ∩ cin 3940  𝒫 cpw 4595   “ cima 5670  ‘cfv 6534  Fincfn 8936  Basecbs 17149  LModclmod 20702  LSpanclspn 20814  LFinGenclfig 42361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-in 3948  df-ss 3958  df-lfig 42362

This theorem is referenced by:  lnrfg  42413