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Theorem lmhmlin 15792
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 2283 . . . . . 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 15784 . . . . 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 5865 . . . . . 6  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
1110fveq2d 5529 . . . . 5  |-  ( a  =  X  ->  ( F `  ( a  .x.  b ) )  =  ( F `  ( X  .x.  b ) ) )
12 oveq1 5865 . . . . 5  |-  ( a  =  X  ->  (
a  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  b )
) )
1311, 12eqeq12d 2297 . . . 4  |-  ( a  =  X  ->  (
( F `  (
a  .x.  b )
)  =  ( a 
.X.  ( F `  b ) )  <->  ( F `  ( X  .x.  b
) )  =  ( X  .X.  ( F `  b ) ) ) )
14 oveq2 5866 . . . . . 6  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
1514fveq2d 5529 . . . . 5  |-  ( b  =  Y  ->  ( F `  ( X  .x.  b ) )  =  ( F `  ( X  .x.  Y ) ) )
16 fveq2 5525 . . . . . 6  |-  ( b  =  Y  ->  ( F `  b )  =  ( F `  Y ) )
1716oveq2d 5874 . . . . 5  |-  ( b  =  Y  ->  ( X  .X.  ( F `  b ) )  =  ( X  .X.  ( F `  Y )
) )
1815, 17eqeq12d 2297 . . . 4  |-  ( b  =  Y  ->  (
( F `  ( X  .x.  b ) )  =  ( X  .X.  ( F `  b ) )  <->  ( F `  ( X  .x.  Y ) )  =  ( X 
.X.  ( F `  Y ) ) ) )
1913, 18rspc2v 2890 . . 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 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  islmhm2  15795  lmhmco  15800  lmhmplusg  15801  lmhmvsca  15802  lmhmf1o  15803  lmhmima  15804  lmhmpreima  15805  reslmhm  15809  reslmhm2  15810  reslmhm2b  15811  lmhmeql  15812  ipass  16549  nmoleub2lem3  18596  nmoleub3  18600  lindfmm  27297  mendassa  27502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-lmhm 15779
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