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Theorem ghmeqker 16861
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 4004 . . . . . 6  |-  {  .0.  }  =  { ( 0g
`  T ) }
43imaeq2i 5177 . . . . 5  |-  ( `' F " {  .0.  } )  =  ( `' F " { ( 0g `  T ) } )
51, 4eqtri 2449 . . . 4  |-  K  =  ( `' F " { ( 0g `  T ) } )
65eleq2i 2498 . . 3  |-  ( ( U  .-  V )  e.  K  <->  ( U  .-  V )  e.  ( `' F " { ( 0g `  T ) } ) )
7 ghmeqker.b . . . . . . 7  |-  B  =  ( Base `  S
)
8 eqid 2420 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
97, 8ghmf 16839 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
10 ffn 5737 . . . . . 6  |-  ( F : B --> ( Base `  T )  ->  F  Fn  B )
119, 10syl 17 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F  Fn  B )
12113ad2ant1 1026 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  F  Fn  B )
13 fniniseg 6009 . . . 4  |-  ( F  Fn  B  ->  (
( U  .-  V
)  e.  ( `' F " { ( 0g `  T ) } )  <->  ( ( U  .-  V )  e.  B  /\  ( F `
 ( U  .-  V ) )  =  ( 0g `  T
) ) ) )
1412, 13syl 17 . . 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 260 . 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 16837 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
17 ghmeqker.m . . . . . 6  |-  .-  =  ( -g `  S )
187, 17grpsubcl 16686 . . . . 5  |-  ( ( S  e.  Grp  /\  U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  e.  B )
1916, 18syl3an1 1297 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( U  .-  V )  e.  B )
2019biantrurd 510 . . 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 2420 . . . . 5  |-  ( -g `  T )  =  (
-g `  T )
227, 17, 21ghmsub 16843 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( ( F `  U ) ( -g `  T ) ( F `
 V ) ) )
2322eqeq1d 2422 . . 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 258 . 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 16838 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
26253ad2ant1 1026 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  T  e.  Grp )
2793ad2ant1 1026 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  F : B --> ( Base `  T
) )
28 simp2 1006 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  U  e.  B )
2927, 28ffvelrnd 6029 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
30 simp3 1007 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  V  e.  B )
3127, 30ffvelrnd 6029 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
32 eqid 2420 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
338, 32, 21grpsubeq0 16692 . . 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 1264 . 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 283 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   {csn 3993   `'ccnv 4844   "cima 4848    Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296   Basecbs 15081   0gc0g 15298   Grpcgrp 16621   -gcsg 16623    GrpHom cghm 16832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-0g 15300  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-grp 16625  df-minusg 16626  df-sbg 16627  df-ghm 16833
This theorem is referenced by:  kerf1hrm  17912  kercvrlsm  35695
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