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Theorem ghmsub 16402
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 16396 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
213ad2ant1 1017 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  S  e.  Grp )
3 simp3 998 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  V  e.  B )
4 ghmsub.b . . . . . 6  |-  B  =  ( Base `  S
)
5 eqid 2457 . . . . . 6  |-  ( invg `  S )  =  ( invg `  S )
64, 5grpinvcl 16222 . . . . 5  |-  ( ( S  e.  Grp  /\  V  e.  B )  ->  ( ( invg `  S ) `  V
)  e.  B )
72, 3, 6syl2anc 661 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( invg `  S ) `  V
)  e.  B )
8 eqid 2457 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
9 eqid 2457 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
104, 8, 9ghmlin 16399 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  (
( invg `  S ) `  V
)  e.  B )  ->  ( F `  ( U ( +g  `  S
) ( ( invg `  S ) `
 V ) ) )  =  ( ( F `  U ) ( +g  `  T
) ( F `  ( ( invg `  S ) `  V
) ) ) )
117, 10syld3an3 1273 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U
( +g  `  S ) ( ( invg `  S ) `  V
) ) )  =  ( ( F `  U ) ( +g  `  T ) ( F `
 ( ( invg `  S ) `
 V ) ) ) )
12 eqid 2457 . . . . . 6  |-  ( invg `  T )  =  ( invg `  T )
134, 5, 12ghminv 16401 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  B )  ->  ( F `  ( ( invg `  S ) `
 V ) )  =  ( ( invg `  T ) `
 ( F `  V ) ) )
14133adant2 1015 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( ( invg `  S ) `
 V ) )  =  ( ( invg `  T ) `
 ( F `  V ) ) )
1514oveq2d 6312 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  U
) ( +g  `  T
) ( F `  ( ( invg `  S ) `  V
) ) )  =  ( ( F `  U ) ( +g  `  T ) ( ( invg `  T
) `  ( F `  V ) ) ) )
1611, 15eqtrd 2498 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U
( +g  `  S ) ( ( invg `  S ) `  V
) ) )  =  ( ( F `  U ) ( +g  `  T ) ( ( invg `  T
) `  ( F `  V ) ) ) )
17 ghmsub.m . . . . 5  |-  .-  =  ( -g `  S )
184, 8, 5, 17grpsubval 16220 . . . 4  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  =  ( U ( +g  `  S
) ( ( invg `  S ) `
 V ) ) )
1918fveq2d 5876 . . 3  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( F `  ( U ( +g  `  S
) ( ( invg `  S ) `
 V ) ) ) )
20193adant1 1014 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( F `  ( U ( +g  `  S
) ( ( invg `  S ) `
 V ) ) ) )
21 eqid 2457 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
224, 21ghmf 16398 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
23 ffvelrn 6030 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  U  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
24 ffvelrn 6030 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
2523, 24anim12dan 837 . . . . 5  |-  ( ( F : B --> ( Base `  T )  /\  ( U  e.  B  /\  V  e.  B )
)  ->  ( ( F `  U )  e.  ( Base `  T
)  /\  ( F `  V )  e.  (
Base `  T )
) )
2622, 25sylan 471 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  ( U  e.  B  /\  V  e.  B )
)  ->  ( ( F `  U )  e.  ( Base `  T
)  /\  ( F `  V )  e.  (
Base `  T )
) )
27263impb 1192 . . 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 16220 . . 3  |-  ( ( ( F `  U
)  e.  ( Base `  T )  /\  ( F `  V )  e.  ( Base `  T
) )  ->  (
( F `  U
) N ( F `
 V ) )  =  ( ( F `
 U ) ( +g  `  T ) ( ( invg `  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 ) ( ( invg `  T ) `  ( F `  V )
) ) )
3116, 20, 303eqtr4d 2508 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( ( F `  U ) N ( F `  V ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   Grpcgrp 16180   invgcminusg 16181   -gcsg 16182    GrpHom cghm 16391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-sbg 16186  df-ghm 16392
This theorem is referenced by:  ghmnsgima  16417  ghmnsgpreima  16418  ghmeqker  16420  ghmf1  16422  evl1subd  18505  ghmcnp  20739  nmods  21377  qqhucn  28134
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