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Theorem ghmid 14899
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 14895 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 eqid 2366 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
3 ghmid.y . . . . . . 7  |-  Y  =  ( 0g `  S
)
42, 3grpidcl 14720 . . . . . 6  |-  ( S  e.  Grp  ->  Y  e.  ( Base `  S
) )
51, 4syl 15 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  Y  e.  ( Base `  S )
)
6 eqid 2366 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
7 eqid 2366 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
82, 6, 7ghmlin 14898 . . . . 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 1280 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( ( F `  Y ) ( +g  `  T
) ( F `  Y ) ) )
102, 6, 3grplid 14722 . . . . . 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 5636 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( Y ( +g  `  S ) Y ) )  =  ( F `
 Y ) )
139, 12eqtr3d 2400 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( ( F `  Y )
( +g  `  T ) ( F `  Y
) )  =  ( F `  Y ) )
14 ghmgrp2 14896 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
15 eqid 2366 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
162, 15ghmf 14897 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
17 ffvelrn 5770 . . . . 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 14727 . . . 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 2371 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1647    e. wcel 1715   -->wf 5354   ` cfv 5358  (class class class)co 5981   Basecbs 13356   +g cplusg 13416   0gc0g 13610   Grpcgrp 14572    GrpHom cghm 14890
This theorem is referenced by:  ghminv  14900  ghmmhm  14903  ghmpreima  14914  ghmf1  14921  lactghmga  14994  srng0  15835  islmhm2  16005  evlslem2  16459  zrh0  16685  chrrhm  16702  zndvds0  16721  ip0l  16757  nmolb2d  18440  nmoi  18450  nmoix  18451  nmoleub  18453  nmoleub2lem2  18812  nmhmcn  18816  evlslem6  19612  evlslem3  19613  dchrptlem2  20727  kerf1hrm  23627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-0g 13614  df-mnd 14577  df-grp 14699  df-ghm 14891
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