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Theorem lmhmlin 16066
Description: A homomorphism of left modules is  K-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlin.k  |-  K  =  (Scalar `  S )
lmhmlin.b  |-  B  =  ( Base `  K
)
lmhmlin.e  |-  E  =  ( Base `  S
)
lmhmlin.m  |-  .x.  =  ( .s `  S )
lmhmlin.n  |-  .X.  =  ( .s `  T )
Assertion
Ref Expression
lmhmlin  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y )
) )

Proof of Theorem lmhmlin
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlin.k . . . . . 6  |-  K  =  (Scalar `  S )
2 eqid 2404 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
3 lmhmlin.b . . . . . 6  |-  B  =  ( Base `  K
)
4 lmhmlin.e . . . . . 6  |-  E  =  ( Base `  S
)
5 lmhmlin.m . . . . . 6  |-  .x.  =  ( .s `  S )
6 lmhmlin.n . . . . . 6  |-  .X.  =  ( .s `  T )
71, 2, 3, 4, 5, 6islmhm 16058 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  <->  ( ( S  e.  LMod  /\  T  e. 
LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  (Scalar `  T )  =  K  /\  A. a  e.  B  A. b  e.  E  ( F `  ( a  .x.  b
) )  =  ( a  .X.  ( F `  b ) ) ) ) )
87simprbi 451 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  ( F  e.  ( S  GrpHom  T )  /\  (Scalar `  T
)  =  K  /\  A. a  e.  B  A. b  e.  E  ( F `  ( a  .x.  b ) )  =  ( a  .X.  ( F `  b )
) ) )
98simp3d 971 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  A. a  e.  B  A. b  e.  E  ( F `  ( a  .x.  b
) )  =  ( a  .X.  ( F `  b ) ) )
10 oveq1 6047 . . . . . 6  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
1110fveq2d 5691 . . . . 5  |-  ( a  =  X  ->  ( F `  ( a  .x.  b ) )  =  ( F `  ( X  .x.  b ) ) )
12 oveq1 6047 . . . . 5  |-  ( a  =  X  ->  (
a  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  b )
) )
1311, 12eqeq12d 2418 . . . 4  |-  ( a  =  X  ->  (
( F `  (
a  .x.  b )
)  =  ( a 
.X.  ( F `  b ) )  <->  ( F `  ( X  .x.  b
) )  =  ( X  .X.  ( F `  b ) ) ) )
14 oveq2 6048 . . . . . 6  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
1514fveq2d 5691 . . . . 5  |-  ( b  =  Y  ->  ( F `  ( X  .x.  b ) )  =  ( F `  ( X  .x.  Y ) ) )
16 fveq2 5687 . . . . . 6  |-  ( b  =  Y  ->  ( F `  b )  =  ( F `  Y ) )
1716oveq2d 6056 . . . . 5  |-  ( b  =  Y  ->  ( X  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  Y )
) )
1815, 17eqeq12d 2418 . . . 4  |-  ( b  =  Y  ->  (
( F `  ( X  .x.  b ) )  =  ( X  .X.  ( F `  b ) )  <->  ( F `  ( X  .x.  Y ) )  =  ( X 
.X.  ( F `  Y ) ) ) )
1913, 18rspc2v 3018 . . 3  |-  ( ( X  e.  B  /\  Y  e.  E )  ->  ( A. a  e.  B  A. b  e.  E  ( F `  ( a  .x.  b
) )  =  ( a  .X.  ( F `  b ) )  -> 
( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y ) ) ) )
209, 19syl5com 28 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y ) ) ) )
21203impib 1151 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   ` cfv 5413  (class class class)co 6040   Basecbs 13424  Scalarcsca 13487   .scvsca 13488    GrpHom cghm 14958   LModclmod 15905   LMHom clmhm 16050
This theorem is referenced by:  islmhm2  16069  lmhmco  16074  lmhmplusg  16075  lmhmvsca  16076  lmhmf1o  16077  lmhmima  16078  lmhmpreima  16079  reslmhm  16083  reslmhm2  16084  reslmhm2b  16085  lmhmeql  16086  ipass  16831  nmoleub2lem3  19076  nmoleub3  19080  lindfmm  27165  mendassa  27370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-lmhm 16053
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