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Theorem ghmlin 16141
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 2441 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 ghmlin.a . . . . . 6  |-  .+  =  ( +g  `  S )
4 ghmlin.b . . . . . 6  |-  .+^  =  ( +g  `  T )
51, 2, 3, 4isghm 16136 . . . . 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 6284 . . . . . 6  |-  ( a  =  U  ->  (
a  .+  b )  =  ( U  .+  b ) )
98fveq2d 5856 . . . . 5  |-  ( a  =  U  ->  ( F `  ( a  .+  b ) )  =  ( F `  ( U  .+  b ) ) )
10 fveq2 5852 . . . . . 6  |-  ( a  =  U  ->  ( F `  a )  =  ( F `  U ) )
1110oveq1d 6292 . . . . 5  |-  ( a  =  U  ->  (
( F `  a
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 b ) ) )
129, 11eqeq12d 2463 . . . 4  |-  ( a  =  U  ->  (
( F `  (
a  .+  b )
)  =  ( ( F `  a ) 
.+^  ( F `  b ) )  <->  ( F `  ( U  .+  b
) )  =  ( ( F `  U
)  .+^  ( F `  b ) ) ) )
13 oveq2 6285 . . . . . 6  |-  ( b  =  V  ->  ( U  .+  b )  =  ( U  .+  V
) )
1413fveq2d 5856 . . . . 5  |-  ( b  =  V  ->  ( F `  ( U  .+  b ) )  =  ( F `  ( U  .+  V ) ) )
15 fveq2 5852 . . . . . 6  |-  ( b  =  V  ->  ( F `  b )  =  ( F `  V ) )
1615oveq2d 6293 . . . . 5  |-  ( b  =  V  ->  (
( F `  U
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )
1714, 16eqeq12d 2463 . . . 4  |-  ( b  =  V  ->  (
( F `  ( U  .+  b ) )  =  ( ( F `
 U )  .+^  ( F `  b ) )  <->  ( F `  ( U  .+  V ) )  =  ( ( F `  U ) 
.+^  ( F `  V ) ) ) )
1812, 17rspc2v 3203 . . 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 1191 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 972    = wceq 1381    e. wcel 1802   A.wral 2791   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569   Grpcgrp 15922    GrpHom cghm 16133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-ghm 16134
This theorem is referenced by:  ghmid  16142  ghminv  16143  ghmsub  16144  ghmmhm  16146  ghmrn  16149  resghm  16152  ghmpreima  16157  ghmnsgima  16159  ghmnsgpreima  16160  ghmf1o  16165  lactghmga  16298  invghm  16711  ghmplusg  16721  srngadd  17374  islmhm2  17552  evlslem1  18052  mpfind  18073  evl1addd  18245  cygznlem3  18475  psgnco  18486  evpmodpmf1o  18499  ipdir  18541  mdetralt  18977  cpmatacl  19084  mat2pmatghm  19098  ghmcnp  20479  ply1rem  22430  dchrptlem2  23405  abliso  27552  rhmopp  27675  qqhghm  27835  qqhrhm  27836  gicabl  31015
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