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Theorem ghmlin 15743
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 2438 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 ghmlin.a . . . . . 6  |-  .+  =  ( +g  `  S )
4 ghmlin.b . . . . . 6  |-  .+^  =  ( +g  `  T )
51, 2, 3, 4isghm 15738 . . . . 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 6093 . . . . . 6  |-  ( a  =  U  ->  (
a  .+  b )  =  ( U  .+  b ) )
98fveq2d 5690 . . . . 5  |-  ( a  =  U  ->  ( F `  ( a  .+  b ) )  =  ( F `  ( U  .+  b ) ) )
10 fveq2 5686 . . . . . 6  |-  ( a  =  U  ->  ( F `  a )  =  ( F `  U ) )
1110oveq1d 6101 . . . . 5  |-  ( a  =  U  ->  (
( F `  a
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 b ) ) )
129, 11eqeq12d 2452 . . . 4  |-  ( a  =  U  ->  (
( F `  (
a  .+  b )
)  =  ( ( F `  a ) 
.+^  ( F `  b ) )  <->  ( F `  ( U  .+  b
) )  =  ( ( F `  U
)  .+^  ( F `  b ) ) ) )
13 oveq2 6094 . . . . . 6  |-  ( b  =  V  ->  ( U  .+  b )  =  ( U  .+  V
) )
1413fveq2d 5690 . . . . 5  |-  ( b  =  V  ->  ( F `  ( U  .+  b ) )  =  ( F `  ( U  .+  V ) ) )
15 fveq2 5686 . . . . . 6  |-  ( b  =  V  ->  ( F `  b )  =  ( F `  V ) )
1615oveq2d 6102 . . . . 5  |-  ( b  =  V  ->  (
( F `  U
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )
1714, 16eqeq12d 2452 . . . 4  |-  ( b  =  V  ->  (
( F `  ( U  .+  b ) )  =  ( ( F `
 U )  .+^  ( F `  b ) )  <->  ( F `  ( U  .+  V ) )  =  ( ( F `  U ) 
.+^  ( F `  V ) ) ) )
1812, 17rspc2v 3074 . . 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 1183 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 965    = wceq 1369    e. wcel 1756   A.wral 2710   -->wf 5409   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   Grpcgrp 15402    GrpHom cghm 15735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-ghm 15736
This theorem is referenced by:  ghmid  15744  ghminv  15745  ghmsub  15746  ghmmhm  15748  ghmrn  15751  resghm  15754  ghmpreima  15759  ghmnsgima  15761  ghmnsgpreima  15762  ghmf1o  15767  lactghmga  15900  invghm  16309  ghmplusg  16319  srngadd  16922  islmhm2  17099  evlslem1  17581  mpfind  17602  evl1addd  17755  cygznlem3  17982  psgnco  17993  evpmodpmf1o  18006  ipdir  18048  mdetralt  18394  ghmcnp  19665  ply1rem  21615  dchrptlem2  22584  abliso  26127  rhmopp  26255  qqhghm  26386  qqhrhm  26387  gicabl  29425
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