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Theorem lmghm 17460
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 2467 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
2 eqid 2467 . . 3  |-  (Scalar `  T )  =  (Scalar `  T )
31, 2lmhmlem 17458 . 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 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282  Scalarcsca 14554    GrpHom cghm 16059   LModclmod 17295   LMHom clmhm 17448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-lmhm 17451
This theorem is referenced by:  lmhmf  17463  islmhm2  17467  lmhmco  17472  lmhmplusg  17473  lmhmvsca  17474  lmhmf1o  17475  lmhmima  17476  lmhmpreima  17477  reslmhm  17481  reslmhm2  17482  reslmhm2b  17483  lmhmeql  17484  lmimgim  17494  ip0l  18438  ipdir  18441  islindf5  18641  isnmhm2  20994  nmoleub2lem  21332  nmoleub2lem2  21334  nmhmcn  21338  kercvrlsm  30633  pwssplit4  30639  mendrng  30746
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