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Theorem nsgid 14663
Description: The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypothesis
Ref Expression
nsgid.z  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
nsgid  |-  ( G  e.  Grp  ->  B  e.  (NrmSGrp `  G )
)

Proof of Theorem nsgid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgid.z . . 3  |-  B  =  ( Base `  G
)
21subgid 14623 . 2  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
3 simp1 955 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  G  e.  Grp )
4 eqid 2283 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
51, 4grpcl 14495 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
6 simp2 956 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  x  e.  B )
7 eqid 2283 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
81, 7grpsubcl 14546 . . . . 5  |-  ( ( G  e.  Grp  /\  ( x ( +g  `  G ) y )  e.  B  /\  x  e.  B )  ->  (
( x ( +g  `  G ) y ) ( -g `  G
) x )  e.  B )
93, 5, 6, 8syl3anc 1182 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( x ( +g  `  G ) y ) ( -g `  G ) x )  e.  B )
1093expb 1152 . . 3  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( x ( +g  `  G ) y ) ( -g `  G
) x )  e.  B )
1110ralrimivva 2635 . 2  |-  ( G  e.  Grp  ->  A. x  e.  B  A. y  e.  B  ( (
x ( +g  `  G
) y ) (
-g `  G )
x )  e.  B
)
121, 4, 7isnsg3 14651 . 2  |-  ( B  e.  (NrmSGrp `  G
)  <->  ( B  e.  (SubGrp `  G )  /\  A. x  e.  B  A. y  e.  B  ( ( x ( +g  `  G ) y ) ( -g `  G ) x )  e.  B ) )
132, 11, 12sylanbrc 645 1  |-  ( G  e.  Grp  ->  B  e.  (NrmSGrp `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615  NrmSGrpcnsg 14616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-ress 13155  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619
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