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Theorem ghmeqker 14709
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 3652 . . . . . 6  |-  {  .0.  }  =  { ( 0g
`  T ) }
43imaeq2i 5010 . . . . 5  |-  ( `' F " {  .0.  } )  =  ( `' F " { ( 0g `  T ) } )
51, 4eqtri 2303 . . . 4  |-  K  =  ( `' F " { ( 0g `  T ) } )
65eleq2i 2347 . . 3  |-  ( ( U  .-  V )  e.  K  <->  ( U  .-  V )  e.  ( `' F " { ( 0g `  T ) } ) )
7 ghmeqker.b . . . . . . 7  |-  B  =  ( Base `  S
)
8 eqid 2283 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
97, 8ghmf 14687 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
10 ffn 5389 . . . . . 6  |-  ( F : B --> ( Base `  T )  ->  F  Fn  B )
119, 10syl 15 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F  Fn  B )
12113ad2ant1 976 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  F  Fn  B )
13 fniniseg 5646 . . . 4  |-  ( F  Fn  B  ->  (
( U  .-  V
)  e.  ( `' F " { ( 0g `  T ) } )  <->  ( ( U  .-  V )  e.  B  /\  ( F `
 ( U  .-  V ) )  =  ( 0g `  T
) ) ) )
1412, 13syl 15 . . 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 248 . 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 14685 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
17 ghmeqker.m . . . . . 6  |-  .-  =  ( -g `  S )
187, 17grpsubcl 14546 . . . . 5  |-  ( ( S  e.  Grp  /\  U  e.  B  /\  V  e.  B )  ->  ( U  .-  V
)  e.  B )
1916, 18syl3an1 1215 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( U  .-  V )  e.  B )
2019biantrurd 494 . . 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 2283 . . . . 5  |-  ( -g `  T )  =  (
-g `  T )
227, 17, 21ghmsub 14691 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  ( U  .-  V ) )  =  ( ( F `  U ) ( -g `  T ) ( F `
 V ) ) )
2322eqeq1d 2291 . . 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 246 . 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 14686 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
26253ad2ant1 976 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  T  e.  Grp )
2793ad2ant1 976 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  F : B --> ( Base `  T
) )
28 simp2 956 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  U  e.  B )
29 ffvelrn 5663 . . . 4  |-  ( ( F : B --> ( Base `  T )  /\  U  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
3027, 28, 29syl2anc 642 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  U )  e.  ( Base `  T
) )
31 simp3 957 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  V  e.  B )
32 ffvelrn 5663 . . . 4  |-  ( ( F : B --> ( Base `  T )  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
3327, 31, 32syl2anc 642 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  ( F `  V )  e.  ( Base `  T
) )
34 eqid 2283 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
358, 34, 21grpsubeq0 14552 . . 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 ) ) )
3626, 30, 33, 35syl3anc 1182 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( ( F `  U ) ( -g `  T ) ( F `
 V ) )  =  ( 0g `  T )  <->  ( F `  U )  =  ( F `  V ) ) )
3715, 24, 363bitrrd 271 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  B  /\  V  e.  B )  ->  (
( F `  U
)  =  ( F `
 V )  <->  ( U  .-  V )  e.  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {csn 3640   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   0gc0g 13400   Grpcgrp 14362   -gcsg 14365    GrpHom cghm 14680
This theorem is referenced by:  kercvrlsm  27181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-ghm 14681
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