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Theorem lmhmlin 15841
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 2316 . . . . . 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 15833 . . . . 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 450 . . . 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 969 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  A. a  e.  B  A. b  e.  E  ( F `  ( a  .x.  b
) )  =  ( a  .X.  ( F `  b ) ) )
10 oveq1 5907 . . . . . 6  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
1110fveq2d 5567 . . . . 5  |-  ( a  =  X  ->  ( F `  ( a  .x.  b ) )  =  ( F `  ( X  .x.  b ) ) )
12 oveq1 5907 . . . . 5  |-  ( a  =  X  ->  (
a  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  b )
) )
1311, 12eqeq12d 2330 . . . 4  |-  ( a  =  X  ->  (
( F `  (
a  .x.  b )
)  =  ( a 
.X.  ( F `  b ) )  <->  ( F `  ( X  .x.  b
) )  =  ( X  .X.  ( F `  b ) ) ) )
14 oveq2 5908 . . . . . 6  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
1514fveq2d 5567 . . . . 5  |-  ( b  =  Y  ->  ( F `  ( X  .x.  b ) )  =  ( F `  ( X  .x.  Y ) ) )
16 fveq2 5563 . . . . . 6  |-  ( b  =  Y  ->  ( F `  b )  =  ( F `  Y ) )
1716oveq2d 5916 . . . . 5  |-  ( b  =  Y  ->  ( X  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  Y )
) )
1815, 17eqeq12d 2330 . . . 4  |-  ( b  =  Y  ->  (
( F `  ( X  .x.  b ) )  =  ( X  .X.  ( F `  b ) )  <->  ( F `  ( X  .x.  Y ) )  =  ( X 
.X.  ( F `  Y ) ) ) )
1913, 18rspc2v 2924 . . 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 26 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y ) ) ) )
21203impib 1149 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   ` cfv 5292  (class class class)co 5900   Basecbs 13195  Scalarcsca 13258   .scvsca 13259    GrpHom cghm 14729   LModclmod 15676   LMHom clmhm 15825
This theorem is referenced by:  islmhm2  15844  lmhmco  15849  lmhmplusg  15850  lmhmvsca  15851  lmhmf1o  15852  lmhmima  15853  lmhmpreima  15854  reslmhm  15858  reslmhm2  15859  reslmhm2b  15860  lmhmeql  15861  ipass  16605  nmoleub2lem3  18649  nmoleub3  18653  lindfmm  26445  mendassa  26650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-lmhm 15828
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