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Theorem ghmid 14975
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 14971 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 eqid 2412 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
3 ghmid.y . . . . . . 7  |-  Y  =  ( 0g `  S
)
42, 3grpidcl 14796 . . . . . 6  |-  ( S  e.  Grp  ->  Y  e.  ( Base `  S
) )
51, 4syl 16 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  Y  e.  ( Base `  S )
)
6 eqid 2412 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
7 eqid 2412 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
82, 6, 7ghmlin 14974 . . . . 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 1281 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( ( F `  Y ) ( +g  `  T
) ( F `  Y ) ) )
102, 6, 3grplid 14798 . . . . . 6  |-  ( ( S  e.  Grp  /\  Y  e.  ( Base `  S ) )  -> 
( Y ( +g  `  S ) Y )  =  Y )
111, 5, 10syl2anc 643 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Y
( +g  `  S ) Y )  =  Y )
1211fveq2d 5699 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( F `
 Y ) )
139, 12eqtr3d 2446 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( ( F `  Y )
( +g  `  T ) ( F `  Y
) )  =  ( F `  Y ) )
14 ghmgrp2 14972 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
15 eqid 2412 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
162, 15ghmf 14973 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1716, 5ffvelrnd 5838 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  e.  (
Base `  T )
)
18 ghmid.z . . . . 5  |-  .0.  =  ( 0g `  T )
1915, 7, 18grpid 14803 . . . 4  |-  ( ( T  e.  Grp  /\  ( F `  Y )  e.  ( Base `  T
) )  ->  (
( ( F `  Y ) ( +g  `  T ) ( F `
 Y ) )  =  ( F `  Y )  <->  .0.  =  ( F `  Y ) ) )
2014, 17, 19syl2anc 643 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( (
( F `  Y
) ( +g  `  T
) ( F `  Y ) )  =  ( F `  Y
)  <->  .0.  =  ( F `  Y )
) )
2113, 20mpbid 202 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  .0.  =  ( F `  Y ) )
2221eqcomd 2417 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492   0gc0g 13686   Grpcgrp 14648    GrpHom cghm 14966
This theorem is referenced by:  ghminv  14976  ghmmhm  14979  ghmpreima  14990  ghmf1  14997  lactghmga  15070  srng0  15911  islmhm2  16077  evlslem2  16531  zrh0  16758  chrrhm  16775  zndvds0  16794  ip0l  16830  nmolb2d  18713  nmoi  18723  nmoix  18724  nmoleub  18726  nmoleub2lem2  19085  nmhmcn  19089  evlslem6  19895  evlslem3  19896  dchrptlem2  21010  kerf1hrm  24223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-0g 13690  df-mnd 14653  df-grp 14775  df-ghm 14967
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