MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntrsubgnsg Structured version   Unicode version

Theorem cntrsubgnsg 15131
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 3341 . . . . . . . 8  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  Z )
5 eqid 2435 . . . . . . . . . 10  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2435 . . . . . . . . . 10  |-  (Cntz `  M )  =  (Cntz `  M )
75, 6cntrval 15110 . . . . . . . . 9  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  (Cntr `  M
)
8 cntrnsg.z . . . . . . . . 9  |-  Z  =  (Cntr `  M )
97, 8eqtr4i 2458 . . . . . . . 8  |-  ( (Cntz `  M ) `  ( Base `  M ) )  =  Z
104, 9syl6eleqr 2526 . . . . . . 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 2435 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1312, 6cntzi 15120 . . . . . . 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 6088 . . . . 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 14941 . . . . . . 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 14937 . . . . . . . 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 3341 . . . . . 6  |-  ( ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  /\  (
x  e.  ( Base `  M )  /\  y  e.  X ) )  -> 
y  e.  ( Base `  M ) )
21 eqid 2435 . . . . . . 7  |-  ( -g `  M )  =  (
-g `  M )
225, 12, 21grppncan 14871 . . . . . 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 2469 . . . 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 2509 . . 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 2790 . 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 14966 . 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 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   -gcsg 14680  SubGrpcsubg 14930  NrmSGrpcnsg 14931  Cntzccntz 15106  Cntrccntr 15107
This theorem is referenced by:  cntrnsg  15132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-nsg 14934  df-cntz 15108  df-cntr 15109
  Copyright terms: Public domain W3C validator