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Theorem lmhmlin 18001
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 2402 . . . . . 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 17993 . . . . 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 462 . . . 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 1011 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  A. a  e.  B  A. b  e.  E  ( F `  ( a  .x.  b
) )  =  ( a  .X.  ( F `  b ) ) )
10 oveq1 6285 . . . . . 6  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
1110fveq2d 5853 . . . . 5  |-  ( a  =  X  ->  ( F `  ( a  .x.  b ) )  =  ( F `  ( X  .x.  b ) ) )
12 oveq1 6285 . . . . 5  |-  ( a  =  X  ->  (
a  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  b )
) )
1311, 12eqeq12d 2424 . . . 4  |-  ( a  =  X  ->  (
( F `  (
a  .x.  b )
)  =  ( a 
.X.  ( F `  b ) )  <->  ( F `  ( X  .x.  b
) )  =  ( X  .X.  ( F `  b ) ) ) )
14 oveq2 6286 . . . . . 6  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
1514fveq2d 5853 . . . . 5  |-  ( b  =  Y  ->  ( F `  ( X  .x.  b ) )  =  ( F `  ( X  .x.  Y ) ) )
16 fveq2 5849 . . . . . 6  |-  ( b  =  Y  ->  ( F `  b )  =  ( F `  Y ) )
1716oveq2d 6294 . . . . 5  |-  ( b  =  Y  ->  ( X  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  Y )
) )
1815, 17eqeq12d 2424 . . . 4  |-  ( b  =  Y  ->  (
( F `  ( X  .x.  b ) )  =  ( X  .X.  ( F `  b ) )  <->  ( F `  ( X  .x.  Y ) )  =  ( X 
.X.  ( F `  Y ) ) ) )
1913, 18rspc2v 3169 . . 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 1195 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   ` cfv 5569  (class class class)co 6278   Basecbs 14841  Scalarcsca 14912   .scvsca 14913    GrpHom cghm 16588   LModclmod 17832   LMHom clmhm 17985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-lmhm 17988
This theorem is referenced by:  islmhm2  18004  lmhmco  18009  lmhmplusg  18010  lmhmvsca  18011  lmhmf1o  18012  lmhmima  18013  lmhmpreima  18014  reslmhm  18018  reslmhm2  18019  reslmhm2b  18020  lmhmeql  18021  ipass  18978  lindfmm  19154  nmoleub2lem3  21890  nmoleub3  21894  mendassa  35507
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