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Theorem nsgid 14679
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 14639 . 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 2296 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
51, 4grpcl 14511 . . . . 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 2296 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
81, 7grpsubcl 14562 . . . . 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 2648 . 2  |-  ( G  e.  Grp  ->  A. x  e.  B  A. y  e.  B  ( (
x ( +g  `  G
) y ) (
-g `  G )
x )  e.  B
)
121, 4, 7isnsg3 14667 . 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631  NrmSGrpcnsg 14632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-ress 13171  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635
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