Theorem fglmod 40814

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 40809	. . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2	1	ssrab3 4011	. 2 ⊢ LFinGen ⊆ LMod
3	2	sseli 3913	1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2108   ∩ cin 3882  𝒫 cpw 4530   “ cima 5583  ‘cfv 6418  Fincfn 8691  Basecbs 16840  LModclmod 20038  LSpanclspn 20148  LFinGenclfig 40808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-lfig 40809

This theorem is referenced by:  lnrfg  40860