MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subg0 Structured version   Unicode version

Theorem subg0 16081
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 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ressplusg 14616 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
43oveqd 6298 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( ( 0g `  H
) ( +g  `  H
) ( 0g `  H ) ) )
51subggrp 16078 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
6 eqid 2443 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
7 eqid 2443 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
86, 7grpidcl 15952 . . . . 5  |-  ( H  e.  Grp  ->  ( 0g `  H )  e.  ( Base `  H
) )
95, 8syl 16 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  H )
)
10 eqid 2443 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
116, 10, 7grplid 15954 . . . 4  |-  ( ( H  e.  Grp  /\  ( 0g `  H )  e.  ( Base `  H
) )  ->  (
( 0g `  H
) ( +g  `  H
) ( 0g `  H ) )  =  ( 0g `  H
) )
125, 9, 11syl2anc 661 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  H ) ( 0g `  H
) )  =  ( 0g `  H ) )
134, 12eqtrd 2484 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( 0g `  H ) )
14 subgrcl 16080 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
15 eqid 2443 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
1615subgss 16076 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
171subgbas 16079 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
189, 17eleqtrrd 2534 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  S
)
1916, 18sseldd 3490 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  G )
)
20 subg0.i . . . 4  |-  .0.  =  ( 0g `  G )
2115, 2, 20grpid 15959 . . 3  |-  ( ( G  e.  Grp  /\  ( 0g `  H )  e.  ( Base `  G
) )  ->  (
( ( 0g `  H ) ( +g  `  G ) ( 0g
`  H ) )  =  ( 0g `  H )  <->  .0.  =  ( 0g `  H ) ) )
2214, 19, 21syl2anc 661 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( (
( 0g `  H
) ( +g  `  G
) ( 0g `  H ) )  =  ( 0g `  H
)  <->  .0.  =  ( 0g `  H ) ) )
2313, 22mpbid 210 1  |-  ( S  e.  (SubGrp `  G
)  ->  .0.  =  ( 0g `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   0gc0g 14714   Grpcgrp 15927  SubGrpcsubg 16069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-subg 16072
This theorem is referenced by:  subginv  16082  subg0cl  16083  subgmulg  16089  subgga  16212  gasubg  16214  sylow2blem2  16515  subgdmdprd  16955  pgpfaclem1  17006  subrg0  17310  abvres  17362  mpl0  17977  gzrngunitlem  18356  prmirredOLD  18401  frlm0  18658  frlmgsumOLD  18674  frlmgsum  18675  subgnm  21020  cphsubrglem  21497  qrng0  23678  suborng  27678  pwssplit4  31010
  Copyright terms: Public domain W3C validator