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Theorem subg0 14878
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 . . . . 5  |-  H  =  ( Gs  S )
2 eqid 2388 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ressplusg 13499 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
43oveqd 6038 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( ( 0g `  H
) ( +g  `  H
) ( 0g `  H ) ) )
51subggrp 14875 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
6 eqid 2388 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
7 eqid 2388 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
86, 7grpidcl 14761 . . . . 5  |-  ( H  e.  Grp  ->  ( 0g `  H )  e.  ( Base `  H
) )
95, 8syl 16 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  H )
)
10 eqid 2388 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
116, 10, 7grplid 14763 . . . 4  |-  ( ( H  e.  Grp  /\  ( 0g `  H )  e.  ( Base `  H
) )  ->  (
( 0g `  H
) ( +g  `  H
) ( 0g `  H ) )  =  ( 0g `  H
) )
125, 9, 11syl2anc 643 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  H ) ( 0g `  H
) )  =  ( 0g `  H ) )
134, 12eqtrd 2420 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( 0g `  H ) )
14 subgrcl 14877 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
15 eqid 2388 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
1615subgss 14873 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
171subgbas 14876 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
189, 17eleqtrrd 2465 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  S
)
1916, 18sseldd 3293 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  G )
)
20 subg0.i . . . 4  |-  .0.  =  ( 0g `  G )
2115, 2, 20grpid 14768 . . 3  |-  ( ( G  e.  Grp  /\  ( 0g `  H )  e.  ( Base `  G
) )  ->  (
( ( 0g `  H ) ( +g  `  G ) ( 0g
`  H ) )  =  ( 0g `  H )  <->  .0.  =  ( 0g `  H ) ) )
2214, 19, 21syl2anc 643 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( (
( 0g `  H
) ( +g  `  G
) ( 0g `  H ) )  =  ( 0g `  H
)  <->  .0.  =  ( 0g `  H ) ) )
2313, 22mpbid 202 1  |-  ( S  e.  (SubGrp `  G
)  ->  .0.  =  ( 0g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   Basecbs 13397   ↾s cress 13398   +g cplusg 13457   0gc0g 13651   Grpcgrp 14613  SubGrpcsubg 14866
This theorem is referenced by:  subginv  14879  subg0cl  14880  subgmulg  14886  subgga  15005  gasubg  15007  sylow2blem2  15183  subgdmdprd  15520  pgpfaclem1  15567  subrg0  15803  abvres  15855  mpl0  16432  gzrngunitlem  16687  prmirred  16699  subgnm  18546  cphsubrglem  19012  qrng0  21183  subofld  24072  pwssplit4  26861  frlm0  26892  frlmgsum  26902
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-mnd 14618  df-grp 14740  df-subg 14869
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