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Theorem ghmlin 16060
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 2460 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 ghmlin.a . . . . . 6  |-  .+  =  ( +g  `  S )
4 ghmlin.b . . . . . 6  |-  .+^  =  ( +g  `  T )
51, 2, 3, 4isghm 16055 . . . . 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 464 . . . 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 463 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  A. a  e.  X  A. b  e.  X  ( F `  ( a  .+  b
) )  =  ( ( F `  a
)  .+^  ( F `  b ) ) )
8 oveq1 6282 . . . . . 6  |-  ( a  =  U  ->  (
a  .+  b )  =  ( U  .+  b ) )
98fveq2d 5861 . . . . 5  |-  ( a  =  U  ->  ( F `  ( a  .+  b ) )  =  ( F `  ( U  .+  b ) ) )
10 fveq2 5857 . . . . . 6  |-  ( a  =  U  ->  ( F `  a )  =  ( F `  U ) )
1110oveq1d 6290 . . . . 5  |-  ( a  =  U  ->  (
( F `  a
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 b ) ) )
129, 11eqeq12d 2482 . . . 4  |-  ( a  =  U  ->  (
( F `  (
a  .+  b )
)  =  ( ( F `  a ) 
.+^  ( F `  b ) )  <->  ( F `  ( U  .+  b
) )  =  ( ( F `  U
)  .+^  ( F `  b ) ) ) )
13 oveq2 6283 . . . . . 6  |-  ( b  =  V  ->  ( U  .+  b )  =  ( U  .+  V
) )
1413fveq2d 5861 . . . . 5  |-  ( b  =  V  ->  ( F `  ( U  .+  b ) )  =  ( F `  ( U  .+  V ) ) )
15 fveq2 5857 . . . . . 6  |-  ( b  =  V  ->  ( F `  b )  =  ( F `  V ) )
1615oveq2d 6291 . . . . 5  |-  ( b  =  V  ->  (
( F `  U
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )
1714, 16eqeq12d 2482 . . . 4  |-  ( b  =  V  ->  (
( F `  ( U  .+  b ) )  =  ( ( F `
 U )  .+^  ( F `  b ) )  <->  ( F `  ( U  .+  V ) )  =  ( ( F `  U ) 
.+^  ( F `  V ) ) ) )
1812, 17rspc2v 3216 . . 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 469 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  ( U  e.  X  /\  V  e.  X )
)  ->  ( F `  ( U  .+  V
) )  =  ( ( F `  U
)  .+^  ( F `  V ) ) )
20193impb 1187 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  X  /\  V  e.  X )  ->  ( F `  ( U  .+  V ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   -->wf 5575   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716    GrpHom cghm 16052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-ghm 16053
This theorem is referenced by:  ghmid  16061  ghminv  16062  ghmsub  16063  ghmmhm  16065  ghmrn  16068  resghm  16071  ghmpreima  16076  ghmnsgima  16078  ghmnsgpreima  16079  ghmf1o  16084  lactghmga  16217  invghm  16628  ghmplusg  16638  srngadd  17282  islmhm2  17460  evlslem1  17948  mpfind  17969  evl1addd  18141  cygznlem3  18368  psgnco  18379  evpmodpmf1o  18392  ipdir  18434  mdetralt  18870  cpmatacl  18977  mat2pmatghm  18991  ghmcnp  20341  ply1rem  22292  dchrptlem2  23261  abliso  27334  rhmopp  27458  qqhghm  27591  qqhrhm  27592  gicabl  30640
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