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Theorem ssnmz 16114
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 16073 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
32sselda 3509 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  X )
4 simpll 753 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
5 subgrcl 16077 . . . . . . . . . . . . 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 759 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  z  e.  S )
97, 8sseldd 3510 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
12 eqid 2467 . . . . . . . . . . . . 13  |-  ( invg `  G )  =  ( invg `  G )
131, 10, 11, 12grplinv 15967 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  z  e.  X )  ->  ( ( ( invg `  G ) `
 z )  .+  z )  =  ( 0g `  G ) )
146, 9, 13syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( invg `  G ) `  z
)  .+  z )  =  ( 0g `  G ) )
1514oveq1d 6310 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( ( invg `  G ) `  z
)  .+  z )  .+  w )  =  ( ( 0g `  G
)  .+  w )
)
1612subginvcl 16081 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  (
( invg `  G ) `  z
)  e.  S )
174, 8, 16syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( invg `  G ) `
 z )  e.  S )
187, 17sseldd 3510 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( invg `  G ) `
 z )  e.  X )
19 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  X )
201, 10grpass 15935 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( ( ( invg `  G ) `
 z )  e.  X  /\  z  e.  X  /\  w  e.  X ) )  -> 
( ( ( ( invg `  G
) `  z )  .+  z )  .+  w
)  =  ( ( ( invg `  G ) `  z
)  .+  ( z  .+  w ) ) )
216, 18, 9, 19, 20syl13anc 1230 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( ( invg `  G ) `  z
)  .+  z )  .+  w )  =  ( ( ( invg `  G ) `  z
)  .+  ( z  .+  w ) ) )
221, 10, 11grplid 15951 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X )  ->  ( ( 0g `  G )  .+  w
)  =  w )
236, 19, 22syl2anc 661 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( 0g `  G )  .+  w )  =  w )
2415, 21, 233eqtr3d 2516 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( invg `  G ) `  z
)  .+  ( z  .+  w ) )  =  w )
25 simpr 461 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( z  .+  w )  e.  S
)
2610subgcl 16082 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (
( invg `  G ) `  z
)  e.  S  /\  ( z  .+  w
)  e.  S )  ->  ( ( ( invg `  G
) `  z )  .+  ( z  .+  w
) )  e.  S
)
274, 17, 25, 26syl3anc 1228 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( invg `  G ) `  z
)  .+  ( z  .+  w ) )  e.  S )
2824, 27eqeltrrd 2556 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  S )
2910subgcl 16082 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  w  e.  S  /\  z  e.  S )  ->  (
w  .+  z )  e.  S )
304, 28, 8, 29syl3anc 1228 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( w  .+  z )  e.  S
)
31 simpll 753 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
32 simplrl 759 . . . . . . . 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 760 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  X )
3531, 32, 3syl2anc 661 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  z  e.  X )
36 eqid 2467 . . . . . . . . . . 11  |-  ( -g `  G )  =  (
-g `  G )
371, 10, 36grppncan 16000 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  z  e.  X )  ->  ( ( w  .+  z ) ( -g `  G ) z )  =  w )
3833, 34, 35, 37syl3anc 1228 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  =  w )
39 simpr 461 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( w  .+  z )  e.  S
)
4036subgsubcl 16083 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (
w  .+  z )  e.  S  /\  z  e.  S )  ->  (
( w  .+  z
) ( -g `  G
) z )  e.  S )
4131, 39, 32, 40syl3anc 1228 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  e.  S
)
4238, 41eqeltrrd 2556 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  S )
4310subgcl 16082 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S  /\  w  e.  S )  ->  (
z  .+  w )  e.  S )
4431, 32, 42, 43syl3anc 1228 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( z  .+  w )  e.  S
)
4530, 44impbida 830 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  ->  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) )
4645anassrs 648 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  /\  w  e.  X )  ->  (
( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
) )
4746ralrimiva 2881 . . . 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 16111 . . . 4  |-  ( z  e.  N  <->  ( z  e.  X  /\  A. w  e.  X  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) ) )
503, 47, 49sylanbrc 664 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  N )
5150ex 434 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( z  e.  S  ->  z  e.  N ) )
5251ssrdv 3515 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821    C_ wss 3481   ` cfv 5594  (class class class)co 6295   Basecbs 14506   +g cplusg 14571   0gc0g 14711   Grpcgrp 15924   invgcminusg 15925   -gcsg 15926  SubGrpcsubg 16066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069
This theorem is referenced by:  nmznsg  16116  sylow3lem6  16523
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