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Theorem subg0 15798
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 2451 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ressplusg 14391 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
43oveqd 6210 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( ( 0g `  H
) ( +g  `  H
) ( 0g `  H ) ) )
51subggrp 15795 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
6 eqid 2451 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
7 eqid 2451 . . . . . 6  |-  ( 0g
`  H )  =  ( 0g `  H
)
86, 7grpidcl 15677 . . . . 5  |-  ( H  e.  Grp  ->  ( 0g `  H )  e.  ( Base `  H
) )
95, 8syl 16 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  H )
)
10 eqid 2451 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
116, 10, 7grplid 15679 . . . 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 2492 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( 0g `  H ) ( +g  `  G ) ( 0g `  H
) )  =  ( 0g `  H ) )
14 subgrcl 15797 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
15 eqid 2451 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
1615subgss 15793 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
171subgbas 15796 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
189, 17eleqtrrd 2542 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  S
)
1916, 18sseldd 3458 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  H )  e.  (
Base `  G )
)
20 subg0.i . . . 4  |-  .0.  =  ( 0g `  G )
2115, 2, 20grpid 15684 . . 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 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   Basecbs 14285   ↾s cress 14286   +g cplusg 14349   0gc0g 14489   Grpcgrp 15521  SubGrpcsubg 15786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-0g 14491  df-mnd 15526  df-grp 15656  df-subg 15789
This theorem is referenced by:  subginv  15799  subg0cl  15800  subgmulg  15806  subgga  15929  gasubg  15931  sylow2blem2  16233  subgdmdprd  16645  pgpfaclem1  16696  subrg0  16987  abvres  17039  mpl0  17636  gzrngunitlem  17995  prmirredOLD  18040  frlm0  18297  frlmgsumOLD  18313  frlmgsum  18314  subgnm  20344  cphsubrglem  20821  qrng0  22996  suborng  26421  pwssplit4  29583
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