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Theorem cntrsubgnsg 15066
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 444 . 2  |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  X  e.  (SubGrp `  M )
)
2 simplr 732 . . . . . . . . 9  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  X  C_  Z )
3 simprr 734 . . . . . . . . 9  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  X )
42, 3sseldd 3292 . . . . . . . 8  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  Z )
5 eqid 2387 . . . . . . . . . 10  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2387 . . . . . . . . . 10  |-  (Cntz `  M )  =  (Cntz `  M )
75, 6cntrval 15045 . . . . . . . . 9  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  (Cntr `  M
)
8 cntrnsg.z . . . . . . . . 9  |-  Z  =  (Cntr `  M )
97, 8eqtr4i 2410 . . . . . . . 8  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  Z
104, 9syl6eleqr 2478 . . . . . . 7  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  ( (Cntz `  M ) `  ( Base `  M ) ) )
11 simprl 733 . . . . . . 7  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  x  e.  ( Base `  M ) )
12 eqid 2387 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1312, 6cntzi 15055 . . . . . . 7  |-  ( ( y  e.  ( (Cntz `  M ) `  ( Base `  M ) )  /\  x  e.  (
Base `  M )
)  ->  ( y
( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) )
1410, 11, 13syl2anc 643 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
1514oveq1d 6035 . . . . 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 14876 . . . . . . 7  |-  ( X  e.  (SubGrp `  M
)  ->  M  e.  Grp )
1716ad2antrr 707 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  M  e.  Grp )
185subgss 14872 . . . . . . . 8  |-  ( X  e.  (SubGrp `  M
)  ->  X  C_  ( Base `  M ) )
1918ad2antrr 707 . . . . . . 7  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  ->  X  C_  ( Base `  M
) )
2019, 3sseldd 3292 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  ( Base `  M ) )
21 eqid 2387 . . . . . . 7  |-  ( -g `  M )  =  (
-g `  M )
225, 12, 21grppncan 14806 . . . . . 6  |-  ( ( M  e.  Grp  /\  y  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  (
( y ( +g  `  M ) x ) ( -g `  M
) x )  =  y )
2317, 20, 11, 22syl3anc 1184 . . . . 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 2421 . . . 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 2461 . . 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 2741 . 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 14901 . 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 646 1  |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  X  e.  (NrmSGrp `  M )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   Grpcgrp 14612   -gcsg 14615  SubGrpcsubg 14865  NrmSGrpcnsg 14866  Cntzccntz 15041  Cntrccntr 15042
This theorem is referenced by:  cntrnsg  15067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-nsg 14869  df-cntz 15043  df-cntr 15044
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