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Theorem ssnmz 14982
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 14945 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
32sselda 3348 . . . 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 14949 . . . . . . . . . . . . 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 3349 . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
12 eqid 2436 . . . . . . . . . . . . 13  |-  ( inv g `  G )  =  ( inv g `  G )
131, 10, 11, 12grplinv 14851 . . . . . . . . . . . 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 6096 . . . . . . . . . 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 14953 . . . . . . . . . . . . 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 3349 . . . . . . . . . . 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 14819 . . . . . . . . . . 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 14835 . . . . . . . . . . 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 2476 . . . . . . . . 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 14954 . . . . . . . . . 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 2511 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  S )
2910subgcl 14954 . . . . . . . 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 2436 . . . . . . . . . . 11  |-  ( -g `  G )  =  (
-g `  G )
371, 10, 36grppncan 14879 . . . . . . . . . 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 14955 . . . . . . . . . 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 2511 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  S )
4310subgcl 14954 . . . . . . . 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 2789 . . . 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 14979 . . . 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 3354 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686   -gcsg 14688  SubGrpcsubg 14938
This theorem is referenced by:  nmznsg  14984  sylow3lem6  15266
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941
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