Theorem fglmod 44279

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 44274	. . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2	1	ssrab3 4010	. 2 ⊢ LFinGen ⊆ LMod
3	2	sseli 3909	1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092  𝒫 cpw 4560   “ cima 5659  ‘cfv 6561  Fincfn 9004  Basecbs 17419  LModclmod 21168  LSpanclspn 21279  LFinGenclfig 44273

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-ss 3901  df-lfig 44274

This theorem is used by:  lnrfg  44324