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Theorem cntrsubgnsg 14832
Description: A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypothesis
Ref Expression
cntrnsg.z  |-  Z  =  (Cntr `  M )
Assertion
Ref Expression
cntrsubgnsg  |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  X  e.  (NrmSGrp `  M )
)

Proof of Theorem cntrsubgnsg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . 2  |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  X  e.  (SubGrp `  M )
)
2 simplr 731 . . . . . . . . 9  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  X  C_  Z )
3 simprr 733 . . . . . . . . 9  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  X )
42, 3sseldd 3194 . . . . . . . 8  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  Z )
5 eqid 2296 . . . . . . . . . 10  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2296 . . . . . . . . . 10  |-  (Cntz `  M )  =  (Cntz `  M )
75, 6cntrval 14811 . . . . . . . . 9  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  (Cntr `  M
)
8 cntrnsg.z . . . . . . . . 9  |-  Z  =  (Cntr `  M )
97, 8eqtr4i 2319 . . . . . . . 8  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  Z
104, 9syl6eleqr 2387 . . . . . . 7  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  ( (Cntz `  M ) `  ( Base `  M ) ) )
11 simprl 732 . . . . . . 7  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  x  e.  ( Base `  M ) )
12 eqid 2296 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1312, 6cntzi 14821 . . . . . . 7  |-  ( ( y  e.  ( (Cntz `  M ) `  ( Base `  M ) )  /\  x  e.  (
Base `  M )
)  ->  ( y
( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) )
1410, 11, 13syl2anc 642 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
1514oveq1d 5889 . . . . 5  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
( ( y ( +g  `  M ) x ) ( -g `  M ) x )  =  ( ( x ( +g  `  M
) y ) (
-g `  M )
x ) )
16 subgrcl 14642 . . . . . . 7  |-  ( X  e.  (SubGrp `  M
)  ->  M  e.  Grp )
1716ad2antrr 706 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  M  e.  Grp )
185subgss 14638 . . . . . . . 8  |-  ( X  e.  (SubGrp `  M
)  ->  X  C_  ( Base `  M ) )
1918ad2antrr 706 . . . . . . 7  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  X  C_  ( Base `  M
) )
2019, 3sseldd 3194 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  ( Base `  M ) )
21 eqid 2296 . . . . . . 7  |-  ( -g `  M )  =  (
-g `  M )
225, 12, 21grppncan 14572 . . . . . 6  |-  ( ( M  e.  Grp  /\  y  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  (
( y ( +g  `  M ) x ) ( -g `  M
) x )  =  y )
2317, 20, 11, 22syl3anc 1182 . . . . 5  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
( ( y ( +g  `  M ) x ) ( -g `  M ) x )  =  y )
2415, 23eqtr3d 2330 . . . 4  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
( ( x ( +g  `  M ) y ) ( -g `  M ) x )  =  y )
2524, 3eqeltrd 2370 . . 3  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
( ( x ( +g  `  M ) y ) ( -g `  M ) x )  e.  X )
2625ralrimivva 2648 . 2  |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  A. x  e.  ( Base `  M
) A. y  e.  X  ( ( x ( +g  `  M
) y ) (
-g `  M )
x )  e.  X
)
275, 12, 21isnsg3 14667 . 2  |-  ( X  e.  (NrmSGrp `  M
)  <->  ( X  e.  (SubGrp `  M )  /\  A. x  e.  (
Base `  M ) A. y  e.  X  ( ( x ( +g  `  M ) y ) ( -g `  M ) x )  e.  X ) )
281, 26, 27sylanbrc 645 1  |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  X  e.  (NrmSGrp `  M )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631  NrmSGrpcnsg 14632  Cntzccntz 14807  Cntrccntr 14808
This theorem is referenced by:  cntrnsg  14833
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-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635  df-cntz 14809  df-cntr 14810
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