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Theorem ghmid 14689
Description: A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmid.y  |-  Y  =  ( 0g `  S
)
ghmid.z  |-  .0.  =  ( 0g `  T )
Assertion
Ref Expression
ghmid  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )

Proof of Theorem ghmid
StepHypRef Expression
1 ghmgrp1 14685 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 eqid 2283 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
3 ghmid.y . . . . . . 7  |-  Y  =  ( 0g `  S
)
42, 3grpidcl 14510 . . . . . 6  |-  ( S  e.  Grp  ->  Y  e.  ( Base `  S
) )
51, 4syl 15 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  Y  e.  ( Base `  S )
)
6 eqid 2283 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
7 eqid 2283 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
82, 6, 7ghmlin 14688 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  Y  e.  ( Base `  S
)  /\  Y  e.  ( Base `  S )
)  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( ( F `  Y ) ( +g  `  T
) ( F `  Y ) ) )
95, 5, 8mpd3an23 1279 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( ( F `  Y ) ( +g  `  T
) ( F `  Y ) ) )
102, 6, 3grplid 14512 . . . . . 6  |-  ( ( S  e.  Grp  /\  Y  e.  ( Base `  S ) )  -> 
( Y ( +g  `  S ) Y )  =  Y )
111, 5, 10syl2anc 642 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Y
( +g  `  S ) Y )  =  Y )
1211fveq2d 5529 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( F `
 Y ) )
139, 12eqtr3d 2317 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( ( F `  Y )
( +g  `  T ) ( F `  Y
) )  =  ( F `  Y ) )
14 ghmgrp2 14686 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
15 eqid 2283 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
162, 15ghmf 14687 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
17 ffvelrn 5663 . . . . 5  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  Y  e.  ( Base `  S
) )  ->  ( F `  Y )  e.  ( Base `  T
) )
1816, 5, 17syl2anc 642 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  e.  (
Base `  T )
)
19 ghmid.z . . . . 5  |-  .0.  =  ( 0g `  T )
2015, 7, 19grpid 14517 . . . 4  |-  ( ( T  e.  Grp  /\  ( F `  Y )  e.  ( Base `  T
) )  ->  (
( ( F `  Y ) ( +g  `  T ) ( F `
 Y ) )  =  ( F `  Y )  <->  .0.  =  ( F `  Y ) ) )
2114, 18, 20syl2anc 642 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( (
( F `  Y
) ( +g  `  T
) ( F `  Y ) )  =  ( F `  Y
)  <->  .0.  =  ( F `  Y )
) )
2213, 21mpbid 201 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  .0.  =  ( F `  Y ) )
2322eqcomd 2288 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362    GrpHom cghm 14680
This theorem is referenced by:  ghminv  14690  ghmmhm  14693  ghmpreima  14704  ghmf1  14711  lactghmga  14784  srng0  15625  islmhm2  15795  evlslem2  16249  zrh0  16468  chrrhm  16485  zndvds0  16504  ip0l  16540  nmolb2d  18227  nmoi  18237  nmoix  18238  nmoleub  18240  nmoleub2lem2  18597  nmhmcn  18601  evlslem6  19397  evlslem3  19398  dchrptlem2  20504
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-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-ghm 14681
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