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Theorem ghmid 15012
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 15008 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 eqid 2436 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
3 ghmid.y . . . . . . 7  |-  Y  =  ( 0g `  S
)
42, 3grpidcl 14833 . . . . . 6  |-  ( S  e.  Grp  ->  Y  e.  ( Base `  S
) )
51, 4syl 16 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  Y  e.  ( Base `  S )
)
6 eqid 2436 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
7 eqid 2436 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
82, 6, 7ghmlin 15011 . . . . 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 14835 . . . . . 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 5732 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( F `
 Y ) )
139, 12eqtr3d 2470 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( ( F `  Y )
( +g  `  T ) ( F `  Y
) )  =  ( F `  Y ) )
14 ghmgrp2 15009 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
15 eqid 2436 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
162, 15ghmf 15010 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1716, 5ffvelrnd 5871 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  e.  (
Base `  T )
)
18 ghmid.z . . . . 5  |-  .0.  =  ( 0g `  T )
1915, 7, 18grpid 14840 . . . 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 2441 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685    GrpHom cghm 15003
This theorem is referenced by:  ghminv  15013  ghmmhm  15016  ghmpreima  15027  ghmf1  15034  lactghmga  15107  srng0  15948  islmhm2  16114  evlslem2  16568  zrh0  16795  chrrhm  16812  zndvds0  16831  ip0l  16867  nmolb2d  18752  nmoi  18762  nmoix  18763  nmoleub  18765  nmoleub2lem2  19124  nmhmcn  19128  evlslem6  19934  evlslem3  19935  dchrptlem2  21049  kerf1hrm  24262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-ghm 15004
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