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Theorem ghmsub 14707
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 14701 . . . . . 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 2296 . . . . . 6  |-  ( inv g `  S )  =  ( inv g `  S )
64, 5grpinvcl 14543 . . . . 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 2296 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
9 eqid 2296 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
104, 8, 9ghmlin 14704 . . . 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 2296 . . . . . 6  |-  ( inv g `  T )  =  ( inv g `  T )
134, 5, 12ghminv 14706 . . . . 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 5890 . . 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 2328 . 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 14541 . . . 4  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  =  ( U ( +g  `  S
) ( ( inv g `  S ) `
 V ) ) )
1918fveq2d 5545 . . 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 2296 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
224, 21ghmf 14703 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
23 ffvelrn 5679 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  U  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
24 ffvelrn 5679 . . . . . 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 14541 . . 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 2338 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 1632    e. wcel 1696   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   inv gcminusg 14379   -gcsg 14381    GrpHom cghm 14696
This theorem is referenced by:  ghmnsgima  14722  ghmnsgpreima  14723  ghmeqker  14725  ghmf1  14727  ghmcnp  17813  nmods  18269  evl1subd  19434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-ghm 14697
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