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Theorem ghmeqker 15794
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 3909 . . . . . 6  |-  {  .0.  }  =  { ( 0g
`  T ) }
43imaeq2i 5188 . . . . 5  |-  ( `' F " {  .0.  } )  =  ( `' F " { ( 0g `  T ) } )
51, 4eqtri 2463 . . . 4  |-  K  =  ( `' F " { ( 0g `  T ) } )
65eleq2i 2507 . . 3  |-  ( ( U  .-  V )  e.  K  <->  ( U  .-  V )  e.  ( `' F " { ( 0g `  T ) } ) )
7 ghmeqker.b . . . . . . 7  |-  B  =  ( Base `  S
)
8 eqid 2443 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
97, 8ghmf 15772 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
10 ffn 5580 . . . . . 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 5845 . . . 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 15770 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
17 ghmeqker.m . . . . . 6  |-  .-  =  ( -g `  S )
187, 17grpsubcl 15627 . . . . 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 2443 . . . . 5  |-  ( -g `  T )  =  (
-g `  T )
227, 17, 21ghmsub 15776 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( ( F `  U ) ( -g `  T ) ( F `
 V ) ) )
2322eqeq1d 2451 . . 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 15771 . . . 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 5865 . . 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 5865 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
32 eqid 2443 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
338, 32, 21grpsubeq0 15633 . . 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 3898   `'ccnv 4860   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   Basecbs 14195   0gc0g 14399   Grpcgrp 15431   -gcsg 15434    GrpHom cghm 15765
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-ghm 15766
This theorem is referenced by:  kerf1hrm  16853  kercvrlsm  29462
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