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Theorem lmhmlin 17234
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 2452 . . . . . 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 17226 . . . . 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 464 . . . 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 1002 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  A. a  e.  B  A. b  e.  E  ( F `  ( a  .x.  b
) )  =  ( a  .X.  ( F `  b ) ) )
10 oveq1 6202 . . . . . 6  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
1110fveq2d 5798 . . . . 5  |-  ( a  =  X  ->  ( F `  ( a  .x.  b ) )  =  ( F `  ( X  .x.  b ) ) )
12 oveq1 6202 . . . . 5  |-  ( a  =  X  ->  (
a  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  b )
) )
1311, 12eqeq12d 2474 . . . 4  |-  ( a  =  X  ->  (
( F `  (
a  .x.  b )
)  =  ( a 
.X.  ( F `  b ) )  <->  ( F `  ( X  .x.  b
) )  =  ( X  .X.  ( F `  b ) ) ) )
14 oveq2 6203 . . . . . 6  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
1514fveq2d 5798 . . . . 5  |-  ( b  =  Y  ->  ( F `  ( X  .x.  b ) )  =  ( F `  ( X  .x.  Y ) ) )
16 fveq2 5794 . . . . . 6  |-  ( b  =  Y  ->  ( F `  b )  =  ( F `  Y ) )
1716oveq2d 6211 . . . . 5  |-  ( b  =  Y  ->  ( X  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  Y )
) )
1815, 17eqeq12d 2474 . . . 4  |-  ( b  =  Y  ->  (
( F `  ( X  .x.  b ) )  =  ( X  .X.  ( F `  b ) )  <->  ( F `  ( X  .x.  Y ) )  =  ( X 
.X.  ( F `  Y ) ) ) )
1913, 18rspc2v 3180 . . 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 30 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y ) ) ) )
21203impib 1186 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   ` cfv 5521  (class class class)co 6195   Basecbs 14287  Scalarcsca 14355   .scvsca 14356    GrpHom cghm 15858   LModclmod 17066   LMHom clmhm 17218
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-lmhm 17221
This theorem is referenced by:  islmhm2  17237  lmhmco  17242  lmhmplusg  17243  lmhmvsca  17244  lmhmf1o  17245  lmhmima  17246  lmhmpreima  17247  reslmhm  17251  reslmhm2  17252  reslmhm2b  17253  lmhmeql  17254  ipass  18194  lindfmm  18376  nmoleub2lem3  20797  nmoleub3  20801  mendassa  29694
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