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Theorem cntrsubgnsg 14816
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 3181 . . . . . . . 8  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  Z )
5 eqid 2283 . . . . . . . . . 10  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2283 . . . . . . . . . 10  |-  (Cntz `  M )  =  (Cntz `  M )
75, 6cntrval 14795 . . . . . . . . 9  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  (Cntr `  M
)
8 cntrnsg.z . . . . . . . . 9  |-  Z  =  (Cntr `  M )
97, 8eqtr4i 2306 . . . . . . . 8  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  Z
104, 9syl6eleqr 2374 . . . . . . 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 2283 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1312, 6cntzi 14805 . . . . . . 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 5873 . . . . 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 14626 . . . . . . 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 14622 . . . . . . . 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 3181 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  ( Base `  M ) )
21 eqid 2283 . . . . . . 7  |-  ( -g `  M )  =  (
-g `  M )
225, 12, 21grppncan 14556 . . . . . 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 2317 . . . 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 2357 . . 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 2635 . 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 14651 . 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615  NrmSGrpcnsg 14616  Cntzccntz 14791  Cntrccntr 14792
This theorem is referenced by:  cntrnsg  14817
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-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619  df-cntz 14793  df-cntr 14794
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