Theorem fglmod 43061

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 43056	. . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2	1	ssrab3 4091	. 2 ⊢ LFinGen ⊆ LMod
3	2	sseli 3990	1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2105   ∩ cin 3961  𝒫 cpw 4604   “ cima 5691  ‘cfv 6562  Fincfn 8983  Basecbs 17244  LModclmod 20874  LSpanclspn 20986  LFinGenclfig 43055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-ss 3979  df-lfig 43056

This theorem is referenced by:  lnrfg  43107