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Theorem ghmlin 16471
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 2454 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 ghmlin.a . . . . . 6  |-  .+  =  ( +g  `  S )
4 ghmlin.b . . . . . 6  |-  .+^  =  ( +g  `  T )
51, 2, 3, 4isghm 16466 . . . . 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 462 . . . 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 461 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  A. a  e.  X  A. b  e.  X  ( F `  ( a  .+  b
) )  =  ( ( F `  a
)  .+^  ( F `  b ) ) )
8 oveq1 6277 . . . . . 6  |-  ( a  =  U  ->  (
a  .+  b )  =  ( U  .+  b ) )
98fveq2d 5852 . . . . 5  |-  ( a  =  U  ->  ( F `  ( a  .+  b ) )  =  ( F `  ( U  .+  b ) ) )
10 fveq2 5848 . . . . . 6  |-  ( a  =  U  ->  ( F `  a )  =  ( F `  U ) )
1110oveq1d 6285 . . . . 5  |-  ( a  =  U  ->  (
( F `  a
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 b ) ) )
129, 11eqeq12d 2476 . . . 4  |-  ( a  =  U  ->  (
( F `  (
a  .+  b )
)  =  ( ( F `  a ) 
.+^  ( F `  b ) )  <->  ( F `  ( U  .+  b
) )  =  ( ( F `  U
)  .+^  ( F `  b ) ) ) )
13 oveq2 6278 . . . . . 6  |-  ( b  =  V  ->  ( U  .+  b )  =  ( U  .+  V
) )
1413fveq2d 5852 . . . . 5  |-  ( b  =  V  ->  ( F `  ( U  .+  b ) )  =  ( F `  ( U  .+  V ) ) )
15 fveq2 5848 . . . . . 6  |-  ( b  =  V  ->  ( F `  b )  =  ( F `  V ) )
1615oveq2d 6286 . . . . 5  |-  ( b  =  V  ->  (
( F `  U
)  .+^  ( F `  b ) )  =  ( ( F `  U )  .+^  ( F `
 V ) ) )
1714, 16eqeq12d 2476 . . . 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 467 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  ( U  e.  X  /\  V  e.  X )
)  ->  ( F `  ( U  .+  V
) )  =  ( ( F `  U
)  .+^  ( F `  V ) ) )
20193impb 1190 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   Grpcgrp 16252    GrpHom cghm 16463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-ghm 16464
This theorem is referenced by:  ghmid  16472  ghminv  16473  ghmsub  16474  ghmmhm  16476  ghmrn  16479  resghm  16482  ghmpreima  16487  ghmnsgima  16489  ghmnsgpreima  16490  ghmf1o  16495  lactghmga  16628  invghm  17041  ghmplusg  17051  srngadd  17701  islmhm2  17879  evlslem1  18379  mpfind  18400  evl1addd  18572  cygznlem3  18781  psgnco  18792  evpmodpmf1o  18805  ipdir  18847  mdetralt  19277  cpmatacl  19384  mat2pmatghm  19398  ghmcnp  20779  ply1rem  22730  dchrptlem2  23738  abliso  27920  rhmopp  28044  qqhghm  28203  qqhrhm  28204  gicabl  31288
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