MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmsub Structured version   Unicode version

Theorem ghmsub 15758
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 15752 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
213ad2ant1 1009 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  S  e.  Grp )
3 simp3 990 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  V  e.  B )
4 ghmsub.b . . . . . 6  |-  B  =  ( Base `  S
)
5 eqid 2443 . . . . . 6  |-  ( invg `  S )  =  ( invg `  S )
64, 5grpinvcl 15586 . . . . 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 2443 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
9 eqid 2443 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
104, 8, 9ghmlin 15755 . . . 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 1263 . . 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 2443 . . . . . 6  |-  ( invg `  T )  =  ( invg `  T )
134, 5, 12ghminv 15757 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  B )  ->  ( F `  ( ( invg `  S ) `
 V ) )  =  ( ( invg `  T ) `
 ( F `  V ) ) )
14133adant2 1007 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( ( invg `  S ) `
 V ) )  =  ( ( invg `  T ) `
 ( F `  V ) ) )
1514oveq2d 6110 . . 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 2475 . 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 15584 . . . 4  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  =  ( U ( +g  `  S
) ( ( invg `  S ) `
 V ) ) )
1918fveq2d 5698 . . 3  |-  ( ( U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( F `  ( U ( +g  `  S
) ( ( invg `  S ) `
 V ) ) ) )
20193adant1 1006 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( F `  ( U ( +g  `  S
) ( ( invg `  S ) `
 V ) ) ) )
21 eqid 2443 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
224, 21ghmf 15754 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
23 ffvelrn 5844 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  U  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
24 ffvelrn 5844 . . . . . 6  |-  ( ( F : B --> ( Base `  T )  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
2523, 24anim12dan 833 . . . . 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 1183 . . 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 15584 . . 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 2485 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 965    = wceq 1369    e. wcel 1756   -->wf 5417   ` cfv 5421  (class class class)co 6094   Basecbs 14177   +g cplusg 14241   Grpcgrp 15413   invgcminusg 15414   -gcsg 15416    GrpHom cghm 15747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-1st 6580  df-2nd 6581  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-sbg 15550  df-ghm 15748
This theorem is referenced by:  ghmnsgima  15773  ghmnsgpreima  15774  ghmeqker  15776  ghmf1  15778  evl1subd  17779  ghmcnp  19688  nmods  20326  qqhucn  26424
  Copyright terms: Public domain W3C validator