Theorem fglmod 43044

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 43039	. . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2	1	ssrab3 4057	. 2 ⊢ LFinGen ⊆ LMod
3	2	sseli 3954	1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2108   ∩ cin 3925  𝒫 cpw 4575   “ cima 5657  ‘cfv 6530  Fincfn 8957  Basecbs 17226  LModclmod 20815  LSpanclspn 20926  LFinGenclfig 43038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-ss 3943  df-lfig 43039

This theorem is referenced by:  lnrfg  43090