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Theorem ghmid 16068
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 16064 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 eqid 2467 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
3 ghmid.y . . . . . . 7  |-  Y  =  ( 0g `  S
)
42, 3grpidcl 15879 . . . . . 6  |-  ( S  e.  Grp  ->  Y  e.  ( Base `  S
) )
51, 4syl 16 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  Y  e.  ( Base `  S )
)
6 eqid 2467 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
7 eqid 2467 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
82, 6, 7ghmlin 16067 . . . . 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 1326 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( ( F `  Y ) ( +g  `  T
) ( F `  Y ) ) )
102, 6, 3grplid 15881 . . . . . 6  |-  ( ( S  e.  Grp  /\  Y  e.  ( Base `  S ) )  -> 
( Y ( +g  `  S ) Y )  =  Y )
111, 5, 10syl2anc 661 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Y
( +g  `  S ) Y )  =  Y )
1211fveq2d 5868 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( F `
 Y ) )
139, 12eqtr3d 2510 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( ( F `  Y )
( +g  `  T ) ( F `  Y
) )  =  ( F `  Y ) )
14 ghmgrp2 16065 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
15 eqid 2467 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
162, 15ghmf 16066 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1716, 5ffvelrnd 6020 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  e.  (
Base `  T )
)
18 ghmid.z . . . . 5  |-  .0.  =  ( 0g `  T )
1915, 7, 18grpid 15886 . . . 4  |-  ( ( T  e.  Grp  /\  ( F `  Y )  e.  ( Base `  T
) )  ->  (
( ( F `  Y ) ( +g  `  T ) ( F `
 Y ) )  =  ( F `  Y )  <->  .0.  =  ( F `  Y ) ) )
2014, 17, 19syl2anc 661 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( (
( F `  Y
) ( +g  `  T
) ( F `  Y ) )  =  ( F `  Y
)  <->  .0.  =  ( F `  Y )
) )
2113, 20mpbid 210 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  .0.  =  ( F `  Y ) )
2221eqcomd 2475 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   Grpcgrp 15723    GrpHom cghm 16059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-0g 14693  df-mnd 15728  df-grp 15858  df-ghm 16060
This theorem is referenced by:  ghminv  16069  ghmmhm  16072  ghmpreima  16083  ghmf1  16090  lactghmga  16224  f1rhm0to0  17172  f1rhm0to0ALT  17173  kerf1hrm  17175  srng0  17292  islmhm2  17467  evlslem2  17951  evlslem6  17952  evlslem6OLD  17953  evlslem3  17954  zrh0  18318  chrrhm  18335  zndvds0  18356  ip0l  18438  0mat2pmat  19004  nmolb2d  20960  nmoi  20970  nmoix  20971  nmoleub  20973  nmoleub2lem2  21334  nmhmcn  21338  dchrptlem2  23268
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