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Theorem ghmsub 15014
Description: Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmsub.b  |-  B  =  ( Base `  S
)
ghmsub.m  |-  .-  =  ( -g `  S )
ghmsub.n  |-  N  =  ( -g `  T
)
Assertion
Ref Expression
ghmsub  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( ( F `  U ) N ( F `  V ) ) )

Proof of Theorem ghmsub
StepHypRef Expression
1 ghmgrp1 15008 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
213ad2ant1 978 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  S  e.  Grp )
3 simp3 959 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  V  e.  B )
4 ghmsub.b . . . . . 6  |-  B  =  ( Base `  S
)
5 eqid 2436 . . . . . 6  |-  ( inv g `  S )  =  ( inv g `  S )
64, 5grpinvcl 14850 . . . . 5  |-  ( ( S  e.  Grp  /\  V  e.  B )  ->  ( ( inv g `  S ) `  V
)  e.  B )
72, 3, 6syl2anc 643 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( inv g `  S ) `  V
)  e.  B )
8 eqid 2436 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
9 eqid 2436 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
104, 8, 9ghmlin 15011 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  (
( inv g `  S ) `  V
)  e.  B )  ->  ( F `  ( U ( +g  `  S
) ( ( inv g `  S ) `
 V ) ) )  =  ( ( F `  U ) ( +g  `  T
) ( F `  ( ( inv g `  S ) `  V
) ) ) )
117, 10syld3an3 1229 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U
( +g  `  S ) ( ( inv g `  S ) `  V
) ) )  =  ( ( F `  U ) ( +g  `  T ) ( F `
 ( ( inv g `  S ) `
 V ) ) ) )
12 eqid 2436 . . . . . 6  |-  ( inv g `  T )  =  ( inv g `  T )
134, 5, 12ghminv 15013 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  B )  ->  ( F `  ( ( inv g `  S ) `
 V ) )  =  ( ( inv g `  T ) `
 ( F `  V ) ) )
14133adant2 976 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( ( inv g `  S ) `
 V ) )  =  ( ( inv g `  T ) `
 ( F `  V ) ) )
1514oveq2d 6097 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  U
) ( +g  `  T
) ( F `  ( ( inv g `  S ) `  V
) ) )  =  ( ( F `  U ) ( +g  `  T ) ( ( inv g `  T
) `  ( F `  V ) ) ) )
1611, 15eqtrd 2468 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U
( +g  `  S ) ( ( inv g `  S ) `  V
) ) )  =  ( ( F `  U ) ( +g  `  T ) ( ( inv g `  T
) `  ( F `  V ) ) ) )
17 ghmsub.m . . . . 5  |-  .-  =  ( -g `  S )
184, 8, 5, 17grpsubval 14848 . . . 4  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  =  ( U ( +g  `  S
) ( ( inv g `  S ) `
 V ) ) )
1918fveq2d 5732 . . 3  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( F `  ( U ( +g  `  S
) ( ( inv g `  S ) `
 V ) ) ) )
20193adant1 975 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( F `  ( U ( +g  `  S
) ( ( inv g `  S ) `
 V ) ) ) )
21 eqid 2436 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
224, 21ghmf 15010 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
23 ffvelrn 5868 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  U  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
24 ffvelrn 5868 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
2523, 24anim12dan 811 . . . . 5  |-  ( ( F : B --> ( Base `  T )  /\  ( U  e.  B  /\  V  e.  B )
)  ->  ( ( F `  U )  e.  ( Base `  T
)  /\  ( F `  V )  e.  (
Base `  T )
) )
2622, 25sylan 458 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  ( U  e.  B  /\  V  e.  B )
)  ->  ( ( F `  U )  e.  ( Base `  T
)  /\  ( F `  V )  e.  (
Base `  T )
) )
27263impb 1149 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  U
)  e.  ( Base `  T )  /\  ( F `  V )  e.  ( Base `  T
) ) )
28 ghmsub.n . . . 4  |-  N  =  ( -g `  T
)
2921, 9, 12, 28grpsubval 14848 . . 3  |-  ( ( ( F `  U
)  e.  ( Base `  T )  /\  ( F `  V )  e.  ( Base `  T
) )  ->  (
( F `  U
) N ( F `
 V ) )  =  ( ( F `
 U ) ( +g  `  T ) ( ( inv g `  T ) `  ( F `  V )
) ) )
3027, 29syl 16 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  U
) N ( F `
 V ) )  =  ( ( F `
 U ) ( +g  `  T ) ( ( inv g `  T ) `  ( F `  V )
) ) )
3116, 20, 303eqtr4d 2478 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( ( F `  U ) N ( F `  V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685   inv gcminusg 14686   -gcsg 14688    GrpHom cghm 15003
This theorem is referenced by:  ghmnsgima  15029  ghmnsgpreima  15030  ghmeqker  15032  ghmf1  15034  ghmcnp  18144  nmods  18778  evl1subd  19955  qqhucn  24376
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-rmo 2713  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-1st 6349  df-2nd 6350  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-ghm 15004
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