Theorem fglmod 37643

Description: Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Assertion

Ref	Expression
fglmod	|- ( M e. LFinGen -> M e. LMod )

Proof of Theorem fglmod

Step	Hyp	Ref	Expression
1		df-lfig 37638	. . 3 |- LFinGen = { a e. LMod | ( Base ` a ) e. ( ( LSpan ` a ) " ( ~P ( Base ` a ) i^i Fin ) ) }
2		ssrab2 3687	. . 3 |- { a e. LMod | ( Base ` a ) e. ( ( LSpan ` a ) " ( ~P ( Base ` a ) i^i Fin ) ) } C_ LMod
3	1, 2	eqsstri 3635	. 2 |- LFinGen C_ LMod
4	3	sseli 3599	1 |- ( M e. LFinGen -> M e. LMod )

Syntax hints:   -> wi 4   e. wcel 1990   {crab 2916   i^i cin 3573   ~Pcpw 4158   "cima 5117   ` cfv 5888   Fincfn 7955   Basecbs 15857   LModclmod 18863   LSpanclspn 18971  LFinGenclfig 37637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-in 3581  df-ss 3588  df-lfig 37638

This theorem is referenced by:  lnrfg  37689