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Theorem fglmod 38162
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
StepHypRef Expression
1 df-lfig 38157 . . 3 LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}
2 ssrab2 3834 . . 3 {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} ⊆ LMod
31, 2eqsstri 3782 . 2 LFinGen ⊆ LMod
43sseli 3746 1 (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  {crab 3064  cin 3720  𝒫 cpw 4295  cima 5252  cfv 6031  Fincfn 8108  Basecbs 16063  LModclmod 19072  LSpanclspn 19183  LFinGenclfig 38156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-in 3728  df-ss 3735  df-lfig 38157
This theorem is referenced by:  lnrfg  38208
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