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Theorem ghmsub 14691
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 14685 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
213ad2ant1 976 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  S  e.  Grp )
3 simp3 957 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  V  e.  B )
4 ghmsub.b . . . . . 6  |-  B  =  ( Base `  S
)
5 eqid 2283 . . . . . 6  |-  ( inv g `  S )  =  ( inv g `  S )
64, 5grpinvcl 14527 . . . . 5  |-  ( ( S  e.  Grp  /\  V  e.  B )  ->  ( ( inv g `  S ) `  V
)  e.  B )
72, 3, 6syl2anc 642 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( inv g `  S ) `  V
)  e.  B )
8 eqid 2283 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
9 eqid 2283 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
104, 8, 9ghmlin 14688 . . . 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 1227 . . 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 2283 . . . . . 6  |-  ( inv g `  T )  =  ( inv g `  T )
134, 5, 12ghminv 14690 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  B )  ->  ( F `  ( ( inv g `  S ) `
 V ) )  =  ( ( inv g `  T ) `
 ( F `  V ) ) )
14133adant2 974 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( ( inv g `  S ) `
 V ) )  =  ( ( inv g `  T ) `
 ( F `  V ) ) )
1514oveq2d 5874 . . 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 2315 . 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 14525 . . . 4  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  =  ( U ( +g  `  S
) ( ( inv g `  S ) `
 V ) ) )
1918fveq2d 5529 . . 3  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( F `  ( U ( +g  `  S
) ( ( inv g `  S ) `
 V ) ) ) )
20193adant1 973 . 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 2283 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
224, 21ghmf 14687 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
23 ffvelrn 5663 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  U  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
24 ffvelrn 5663 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
2523, 24anim12dan 810 . . . . 5  |-  ( ( F : B --> ( Base `  T )  /\  ( U  e.  B  /\  V  e.  B )
)  ->  ( ( F `  U )  e.  ( Base `  T
)  /\  ( F `  V )  e.  (
Base `  T )
) )
2622, 25sylan 457 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  ( U  e.  B  /\  V  e.  B )
)  ->  ( ( F `  U )  e.  ( Base `  T
)  /\  ( F `  V )  e.  (
Base `  T )
) )
27263impb 1147 . . 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 14525 . . 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 15 . 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 2325 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365    GrpHom cghm 14680
This theorem is referenced by:  ghmnsgima  14706  ghmnsgpreima  14707  ghmeqker  14709  ghmf1  14711  ghmcnp  17797  nmods  18253  evl1subd  19418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-ghm 14681
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