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Theorem ghmlin 15011
Description: A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmlin.x  |-  X  =  ( Base `  S
)
ghmlin.a  |-  .+  =  ( +g  `  S )
ghmlin.b  |-  .+^  =  ( +g  `  T )
Assertion
Ref Expression
ghmlin  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  X  /\  V  e.  X )  ->  ( F `  ( U  .+  V ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )

Proof of Theorem ghmlin
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmlin.x . . . . . 6  |-  X  =  ( Base `  S
)
2 eqid 2436 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 ghmlin.a . . . . . 6  |-  .+  =  ( +g  `  S )
4 ghmlin.b . . . . . 6  |-  .+^  =  ( +g  `  T )
51, 2, 3, 4isghm 15006 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  <->  ( ( S  e.  Grp  /\  T  e.  Grp )  /\  ( F : X --> ( Base `  T )  /\  A. a  e.  X  A. b  e.  X  ( F `  ( a  .+  b ) )  =  ( ( F `  a )  .+^  ( F `
 b ) ) ) ) )
65simprbi 451 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F : X --> ( Base `  T
)  /\  A. a  e.  X  A. b  e.  X  ( F `  ( a  .+  b
) )  =  ( ( F `  a
)  .+^  ( F `  b ) ) ) )
76simprd 450 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  A. a  e.  X  A. b  e.  X  ( F `  ( a  .+  b
) )  =  ( ( F `  a
)  .+^  ( F `  b ) ) )
8 oveq1 6088 . . . . . 6  |-  ( a  =  U  ->  (
a  .+  b )  =  ( U  .+  b ) )
98fveq2d 5732 . . . . 5  |-  ( a  =  U  ->  ( F `  ( a  .+  b ) )  =  ( F `  ( U  .+  b ) ) )
10 fveq2 5728 . . . . . 6  |-  ( a  =  U  ->  ( F `  a )  =  ( F `  U ) )
1110oveq1d 6096 . . . . 5  |-  ( a  =  U  ->  (
( F `  a
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 b ) ) )
129, 11eqeq12d 2450 . . . 4  |-  ( a  =  U  ->  (
( F `  (
a  .+  b )
)  =  ( ( F `  a ) 
.+^  ( F `  b ) )  <->  ( F `  ( U  .+  b
) )  =  ( ( F `  U
)  .+^  ( F `  b ) ) ) )
13 oveq2 6089 . . . . . 6  |-  ( b  =  V  ->  ( U  .+  b )  =  ( U  .+  V
) )
1413fveq2d 5732 . . . . 5  |-  ( b  =  V  ->  ( F `  ( U  .+  b ) )  =  ( F `  ( U  .+  V ) ) )
15 fveq2 5728 . . . . . 6  |-  ( b  =  V  ->  ( F `  b )  =  ( F `  V ) )
1615oveq2d 6097 . . . . 5  |-  ( b  =  V  ->  (
( F `  U
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )
1714, 16eqeq12d 2450 . . . 4  |-  ( b  =  V  ->  (
( F `  ( U  .+  b ) )  =  ( ( F `
 U )  .+^  ( F `  b ) )  <->  ( F `  ( U  .+  V ) )  =  ( ( F `  U ) 
.+^  ( F `  V ) ) ) )
1812, 17rspc2v 3058 . . 3  |-  ( ( U  e.  X  /\  V  e.  X )  ->  ( A. a  e.  X  A. b  e.  X  ( F `  ( a  .+  b
) )  =  ( ( F `  a
)  .+^  ( F `  b ) )  -> 
( F `  ( U  .+  V ) )  =  ( ( F `
 U )  .+^  ( F `  V ) ) ) )
197, 18mpan9 456 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  ( U  e.  X  /\  V  e.  X )
)  ->  ( F `  ( U  .+  V
) )  =  ( ( F `  U
)  .+^  ( F `  V ) ) )
20193impb 1149 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  X  /\  V  e.  X )  ->  ( F `  ( U  .+  V ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685    GrpHom cghm 15003
This theorem is referenced by:  ghmid  15012  ghminv  15013  ghmsub  15014  ghmmhm  15016  ghmrn  15019  resghm  15022  ghmpreima  15027  ghmnsgima  15029  ghmnsgpreima  15030  ghmf1o  15035  lactghmga  15107  invghm  15453  ghmplusg  15461  srngadd  15945  islmhm2  16114  cygznlem3  16850  ipdir  16870  ghmcnp  18144  evlslem1  19936  evl1addd  19954  mpfind  19965  ply1rem  20086  dchrptlem2  21049  rhmopp  24257  qqhghm  24372  qqhrhm  24373  gicabl  27240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-ghm 15004
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