MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmghm Structured version   Unicode version

Theorem lmghm 17220
Description: A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Assertion
Ref Expression
lmghm  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )

Proof of Theorem lmghm
StepHypRef Expression
1 eqid 2451 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
2 eqid 2451 . . 3  |-  (Scalar `  T )  =  (Scalar `  T )
31, 2lmhmlem 17218 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  (Scalar `  T )  =  (Scalar `  S ) ) ) )
4 simprl 755 . 2  |-  ( ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  (Scalar `  T )  =  (Scalar `  S )
) )  ->  F  e.  ( S  GrpHom  T ) )
53, 4syl 16 1  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192  Scalarcsca 14345    GrpHom cghm 15848   LModclmod 17056   LMHom clmhm 17208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-lmhm 17211
This theorem is referenced by:  lmhmf  17223  islmhm2  17227  lmhmco  17232  lmhmplusg  17233  lmhmvsca  17234  lmhmf1o  17235  lmhmima  17236  lmhmpreima  17237  reslmhm  17241  reslmhm2  17242  reslmhm2b  17243  lmhmeql  17244  lmimgim  17254  ip0l  18176  ipdir  18179  islindf5  18379  isnmhm2  20449  nmoleub2lem  20787  nmoleub2lem2  20789  nmhmcn  20793  kercvrlsm  29576  pwssplit4  29582  mendrng  29689
  Copyright terms: Public domain W3C validator