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Theorem ghmeqker 15762
Description: Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmeqker.b  |-  B  =  ( Base `  S
)
ghmeqker.z  |-  .0.  =  ( 0g `  T )
ghmeqker.k  |-  K  =  ( `' F " {  .0.  } )
ghmeqker.m  |-  .-  =  ( -g `  S )
Assertion
Ref Expression
ghmeqker  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  U
)  =  ( F `
 V )  <->  ( U  .-  V )  e.  K
) )

Proof of Theorem ghmeqker
StepHypRef Expression
1 ghmeqker.k . . . . 5  |-  K  =  ( `' F " {  .0.  } )
2 ghmeqker.z . . . . . . 7  |-  .0.  =  ( 0g `  T )
32sneqi 3881 . . . . . 6  |-  {  .0.  }  =  { ( 0g
`  T ) }
43imaeq2i 5160 . . . . 5  |-  ( `' F " {  .0.  } )  =  ( `' F " { ( 0g `  T ) } )
51, 4eqtri 2457 . . . 4  |-  K  =  ( `' F " { ( 0g `  T ) } )
65eleq2i 2501 . . 3  |-  ( ( U  .-  V )  e.  K  <->  ( U  .-  V )  e.  ( `' F " { ( 0g `  T ) } ) )
7 ghmeqker.b . . . . . . 7  |-  B  =  ( Base `  S
)
8 eqid 2437 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
97, 8ghmf 15740 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
10 ffn 5552 . . . . . 6  |-  ( F : B --> ( Base `  T )  ->  F  Fn  B )
119, 10syl 16 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F  Fn  B )
12113ad2ant1 1009 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  F  Fn  B )
13 fniniseg 5817 . . . 4  |-  ( F  Fn  B  ->  (
( U  .-  V
)  e.  ( `' F " { ( 0g `  T ) } )  <->  ( ( U  .-  V )  e.  B  /\  ( F `
 ( U  .-  V ) )  =  ( 0g `  T
) ) ) )
1412, 13syl 16 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( U  .-  V
)  e.  ( `' F " { ( 0g `  T ) } )  <->  ( ( U  .-  V )  e.  B  /\  ( F `
 ( U  .-  V ) )  =  ( 0g `  T
) ) ) )
156, 14syl5bb 257 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( U  .-  V
)  e.  K  <->  ( ( U  .-  V )  e.  B  /\  ( F `
 ( U  .-  V ) )  =  ( 0g `  T
) ) ) )
16 ghmgrp1 15738 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
17 ghmeqker.m . . . . . 6  |-  .-  =  ( -g `  S )
187, 17grpsubcl 15595 . . . . 5  |-  ( ( S  e.  Grp  /\  U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  e.  B )
1916, 18syl3an1 1251 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( U  .-  V )  e.  B )
2019biantrurd 508 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  ( U  .-  V ) )  =  ( 0g `  T )  <->  ( ( U  .-  V )  e.  B  /\  ( F `
 ( U  .-  V ) )  =  ( 0g `  T
) ) ) )
21 eqid 2437 . . . . 5  |-  ( -g `  T )  =  (
-g `  T )
227, 17, 21ghmsub 15744 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( ( F `  U ) ( -g `  T ) ( F `
 V ) ) )
2322eqeq1d 2445 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  ( U  .-  V ) )  =  ( 0g `  T )  <->  ( ( F `  U )
( -g `  T ) ( F `  V
) )  =  ( 0g `  T ) ) )
2420, 23bitr3d 255 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( ( U  .-  V )  e.  B  /\  ( F `  ( U  .-  V ) )  =  ( 0g `  T ) )  <->  ( ( F `  U )
( -g `  T ) ( F `  V
) )  =  ( 0g `  T ) ) )
25 ghmgrp2 15739 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
26253ad2ant1 1009 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  T  e.  Grp )
2793ad2ant1 1009 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  F : B --> ( Base `  T
) )
28 simp2 989 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  U  e.  B )
2927, 28ffvelrnd 5837 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
30 simp3 990 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  V  e.  B )
3127, 30ffvelrnd 5837 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
32 eqid 2437 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
338, 32, 21grpsubeq0 15601 . . 3  |-  ( ( T  e.  Grp  /\  ( F `  U )  e.  ( Base `  T
)  /\  ( F `  V )  e.  (
Base `  T )
)  ->  ( (
( F `  U
) ( -g `  T
) ( F `  V ) )  =  ( 0g `  T
)  <->  ( F `  U )  =  ( F `  V ) ) )
3426, 29, 31, 33syl3anc 1218 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( ( F `  U ) ( -g `  T ) ( F `
 V ) )  =  ( 0g `  T )  <->  ( F `  U )  =  ( F `  V ) ) )
3515, 24, 343bitrrd 280 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  U
)  =  ( F `
 V )  <->  ( U  .-  V )  e.  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {csn 3870   `'ccnv 4831   "cima 4835    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   Basecbs 14166   0gc0g 14370   Grpcgrp 15402   -gcsg 15405    GrpHom cghm 15733
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
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 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-0g 14372  df-mnd 15407  df-grp 15534  df-minusg 15535  df-sbg 15536  df-ghm 15734
This theorem is referenced by:  kerf1hrm  26239  kercvrlsm  29379
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