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Theorem subg0 12971
Description: A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
subg0.h  |-  H  =  ( Gs  S )
subg0.i  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
subg0  |-  ( S  e.  (SubGrp `  G
)  ->  .0.  =  ( 0g `  H ) )

Proof of Theorem subg0
StepHypRef Expression
1 subg0.h . . . . . 6  |-  H  =  ( Gs  S )
21a1i 9 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  H  =  ( Gs  S ) )
3 eqid 2177 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
43a1i 9 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  G ) )
5 id 19 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
6 subgrcl 12970 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
72, 4, 5, 6ressplusgd 12579 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
87oveqd 5889 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( ( 0g `  H
) ( +g  `  H
) ( 0g `  H ) ) )
91subggrp 12968 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
10 eqid 2177 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
11 eqid 2177 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
1210, 11grpidcl 12836 . . . . 5  |-  ( H  e.  Grp  ->  ( 0g `  H )  e.  ( Base `  H
) )
139, 12syl 14 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  H )
)
14 eqid 2177 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
1510, 14, 11grplid 12838 . . . 4  |-  ( ( H  e.  Grp  /\  ( 0g `  H )  e.  ( Base `  H
) )  ->  (
( 0g `  H
) ( +g  `  H
) ( 0g `  H ) )  =  ( 0g `  H
) )
169, 13, 15syl2anc 411 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  H ) ( 0g `  H
) )  =  ( 0g `  H ) )
178, 16eqtrd 2210 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( 0g `  H ) )
18 eqid 2177 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
1918subgss 12965 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
201subgbas 12969 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
2113, 20eleqtrrd 2257 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  S
)
2219, 21sseldd 3156 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  G )
)
23 subg0.i . . . 4  |-  .0.  =  ( 0g `  G )
2418, 3, 23grpid 12844 . . 3  |-  ( ( G  e.  Grp  /\  ( 0g `  H )  e.  ( Base `  G
) )  ->  (
( ( 0g `  H ) ( +g  `  G ) ( 0g
`  H ) )  =  ( 0g `  H )  <->  .0.  =  ( 0g `  H ) ) )
256, 22, 24syl2anc 411 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( (
( 0g `  H
) ( +g  `  G
) ( 0g `  H ) )  =  ( 0g `  H
)  <->  .0.  =  ( 0g `  H ) ) )
2617, 25mpbid 147 1  |-  ( S  e.  (SubGrp `  G
)  ->  .0.  =  ( 0g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   +g cplusg 12528   0gc0g 12693   Grpcgrp 12809  SubGrpcsubg 12958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-subg 12961
This theorem is referenced by:  subginv  12972  subg0cl  12973  subgmulg  12979  subrg0  13287
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