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Theorem ssnmz 14937
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 14900 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
32sselda 3308 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  X )
4 simpll 731 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
5 subgrcl 14904 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
64, 5syl 16 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  G  e.  Grp )
74, 2syl 16 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  S  C_  X
)
8 simplrl 737 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  z  e.  S )
97, 8sseldd 3309 . . . . . . . . . . . 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 2404 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
12 eqid 2404 . . . . . . . . . . . . 13  |-  ( inv g `  G )  =  ( inv g `  G )
131, 10, 11, 12grplinv 14806 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  z  e.  X )  ->  ( ( ( inv g `  G ) `
 z )  .+  z )  =  ( 0g `  G ) )
146, 9, 13syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( inv g `  G ) `  z
)  .+  z )  =  ( 0g `  G ) )
1514oveq1d 6055 . . . . . . . . . 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 14908 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  (
( inv g `  G ) `  z
)  e.  S )
174, 8, 16syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( inv g `  G ) `
 z )  e.  S )
187, 17sseldd 3309 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( inv g `  G ) `
 z )  e.  X )
19 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  X )
201, 10grpass 14774 . . . . . . . . . . 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 1186 . . . . . . . . . 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 14790 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X )  ->  ( ( 0g `  G )  .+  w
)  =  w )
236, 19, 22syl2anc 643 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( 0g `  G )  .+  w )  =  w )
2415, 21, 233eqtr3d 2444 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( inv g `  G ) `  z
)  .+  ( z  .+  w ) )  =  w )
25 simpr 448 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( z  .+  w )  e.  S
)
2610subgcl 14909 . . . . . . . . . 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 1184 . . . . . . . . 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 2479 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  S )
2910subgcl 14909 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  w  e.  S  /\  z  e.  S )  ->  (
w  .+  z )  e.  S )
304, 28, 8, 29syl3anc 1184 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( w  .+  z )  e.  S
)
31 simpll 731 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
32 simplrl 737 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  z  e.  S )
3331, 5syl 16 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  G  e.  Grp )
34 simplrr 738 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  X )
3531, 32, 3syl2anc 643 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  z  e.  X )
36 eqid 2404 . . . . . . . . . . 11  |-  ( -g `  G )  =  (
-g `  G )
371, 10, 36grppncan 14834 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  z  e.  X )  ->  ( ( w  .+  z ) ( -g `  G ) z )  =  w )
3833, 34, 35, 37syl3anc 1184 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  =  w )
39 simpr 448 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( w  .+  z )  e.  S
)
4036subgsubcl 14910 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (
w  .+  z )  e.  S  /\  z  e.  S )  ->  (
( w  .+  z
) ( -g `  G
) z )  e.  S )
4131, 39, 32, 40syl3anc 1184 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  e.  S
)
4238, 41eqeltrrd 2479 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  S )
4310subgcl 14909 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S  /\  w  e.  S )  ->  (
z  .+  w )  e.  S )
4431, 32, 42, 43syl3anc 1184 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( z  .+  w )  e.  S
)
4530, 44impbida 806 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  ->  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) )
4645anassrs 630 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  /\  w  e.  X )  ->  (
( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
) )
4746ralrimiva 2749 . . . 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 14934 . . . 4  |-  ( z  e.  N  <->  ( z  e.  X  /\  A. w  e.  X  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) ) )
503, 47, 49sylanbrc 646 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  N )
5150ex 424 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( z  e.  S  ->  z  e.  N ) )
5251ssrdv 3314 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    C_ wss 3280   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641   -gcsg 14643  SubGrpcsubg 14893
This theorem is referenced by:  nmznsg  14939  sylow3lem6  15221
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896
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