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Theorem ssnmz 14675
Description: A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
elnmz.1  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }
nmzsubg.2  |-  X  =  ( Base `  G
)
nmzsubg.3  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
ssnmz  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
Distinct variable groups:    x, y, G    x, S, y    x,  .+ , y    x, X, y
Allowed substitution hints:    N( x, y)

Proof of Theorem ssnmz
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmzsubg.2 . . . . . 6  |-  X  =  ( Base `  G
)
21subgss 14638 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
32sselda 3193 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  X )
4 simpll 730 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
5 subgrcl 14642 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
64, 5syl 15 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  G  e.  Grp )
74, 2syl 15 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  S  C_  X
)
8 simplrl 736 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  z  e.  S )
97, 8sseldd 3194 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  z  e.  X )
10 nmzsubg.3 . . . . . . . . . . . . 13  |-  .+  =  ( +g  `  G )
11 eqid 2296 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
12 eqid 2296 . . . . . . . . . . . . 13  |-  ( inv g `  G )  =  ( inv g `  G )
131, 10, 11, 12grplinv 14544 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  z  e.  X )  ->  ( ( ( inv g `  G ) `
 z )  .+  z )  =  ( 0g `  G ) )
146, 9, 13syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( inv g `  G ) `  z
)  .+  z )  =  ( 0g `  G ) )
1514oveq1d 5889 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( ( inv g `  G ) `  z
)  .+  z )  .+  w )  =  ( ( 0g `  G
)  .+  w )
)
1612subginvcl 14646 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  (
( inv g `  G ) `  z
)  e.  S )
174, 8, 16syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( inv g `  G ) `
 z )  e.  S )
187, 17sseldd 3194 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( inv g `  G ) `
 z )  e.  X )
19 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  X )
201, 10grpass 14512 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( ( ( inv g `  G ) `
 z )  e.  X  /\  z  e.  X  /\  w  e.  X ) )  -> 
( ( ( ( inv g `  G
) `  z )  .+  z )  .+  w
)  =  ( ( ( inv g `  G ) `  z
)  .+  ( z  .+  w ) ) )
216, 18, 9, 19, 20syl13anc 1184 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( ( inv g `  G ) `  z
)  .+  z )  .+  w )  =  ( ( ( inv g `  G ) `  z
)  .+  ( z  .+  w ) ) )
221, 10, 11grplid 14528 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X )  ->  ( ( 0g `  G )  .+  w
)  =  w )
236, 19, 22syl2anc 642 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( 0g `  G )  .+  w )  =  w )
2415, 21, 233eqtr3d 2336 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( inv g `  G ) `  z
)  .+  ( z  .+  w ) )  =  w )
25 simpr 447 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( z  .+  w )  e.  S
)
2610subgcl 14647 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (
( inv g `  G ) `  z
)  e.  S  /\  ( z  .+  w
)  e.  S )  ->  ( ( ( inv g `  G
) `  z )  .+  ( z  .+  w
) )  e.  S
)
274, 17, 25, 26syl3anc 1182 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( inv g `  G ) `  z
)  .+  ( z  .+  w ) )  e.  S )
2824, 27eqeltrrd 2371 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  S )
2910subgcl 14647 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  w  e.  S  /\  z  e.  S )  ->  (
w  .+  z )  e.  S )
304, 28, 8, 29syl3anc 1182 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( w  .+  z )  e.  S
)
31 simpll 730 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
32 simplrl 736 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  z  e.  S )
3331, 5syl 15 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  G  e.  Grp )
34 simplrr 737 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  X )
3531, 32, 3syl2anc 642 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  z  e.  X )
36 eqid 2296 . . . . . . . . . . 11  |-  ( -g `  G )  =  (
-g `  G )
371, 10, 36grppncan 14572 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  z  e.  X )  ->  ( ( w  .+  z ) ( -g `  G ) z )  =  w )
3833, 34, 35, 37syl3anc 1182 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  =  w )
39 simpr 447 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( w  .+  z )  e.  S
)
4036subgsubcl 14648 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (
w  .+  z )  e.  S  /\  z  e.  S )  ->  (
( w  .+  z
) ( -g `  G
) z )  e.  S )
4131, 39, 32, 40syl3anc 1182 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  e.  S
)
4238, 41eqeltrrd 2371 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  S )
4310subgcl 14647 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S  /\  w  e.  S )  ->  (
z  .+  w )  e.  S )
4431, 32, 42, 43syl3anc 1182 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( z  .+  w )  e.  S
)
4530, 44impbida 805 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  ->  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) )
4645anassrs 629 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  /\  w  e.  X )  ->  (
( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
) )
4746ralrimiva 2639 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  A. w  e.  X  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) )
48 elnmz.1 . . . . 5  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }
4948elnmz 14672 . . . 4  |-  ( z  e.  N  <->  ( z  e.  X  /\  A. w  e.  X  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) ) )
503, 47, 49sylanbrc 645 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  N )
5150ex 423 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( z  e.  S  ->  z  e.  N ) )
5251ssrdv 3198 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   -gcsg 14381  SubGrpcsubg 14631
This theorem is referenced by:  nmznsg  14677  sylow3lem6  14959
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634
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