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Theorem ssnmz 13002
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 12965 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
32sselda 3155 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  X )
4 simpll 527 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
5 subgrcl 12970 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
64, 5syl 14 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  G  e.  Grp )
74, 2syl 14 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  S  C_  X
)
8 simplrl 535 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  z  e.  S )
97, 8sseldd 3156 . . . . . . . . . . . 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 2177 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
12 eqid 2177 . . . . . . . . . . . . 13  |-  ( invg `  G )  =  ( invg `  G )
131, 10, 11, 12grplinv 12854 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  z  e.  X )  ->  ( ( ( invg `  G ) `
 z )  .+  z )  =  ( 0g `  G ) )
146, 9, 13syl2anc 411 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( invg `  G ) `  z
)  .+  z )  =  ( 0g `  G ) )
1514oveq1d 5887 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( ( invg `  G ) `  z
)  .+  z )  .+  w )  =  ( ( 0g `  G
)  .+  w )
)
1612subginvcl 12974 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  (
( invg `  G ) `  z
)  e.  S )
174, 8, 16syl2anc 411 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( invg `  G ) `
 z )  e.  S )
187, 17sseldd 3156 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( invg `  G ) `
 z )  e.  X )
19 simplrr 536 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  X )
201, 10grpass 12818 . . . . . . . . . . 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 1240 . . . . . . . . . 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 12838 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X )  ->  ( ( 0g `  G )  .+  w
)  =  w )
236, 19, 22syl2anc 411 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( ( 0g `  G )  .+  w )  =  w )
2415, 21, 233eqtr3d 2218 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( invg `  G ) `  z
)  .+  ( z  .+  w ) )  =  w )
25 simpr 110 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( z  .+  w )  e.  S
)
2610subgcl 12975 . . . . . . . . . 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 1238 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( (
( invg `  G ) `  z
)  .+  ( z  .+  w ) )  e.  S )
2824, 27eqeltrrd 2255 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  w  e.  S )
2910subgcl 12975 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  w  e.  S  /\  z  e.  S )  ->  (
w  .+  z )  e.  S )
304, 28, 8, 29syl3anc 1238 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( z  .+  w )  e.  S
)  ->  ( w  .+  z )  e.  S
)
31 simpll 527 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  S  e.  (SubGrp `  G ) )
32 simplrl 535 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  z  e.  S )
3331, 5syl 14 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  G  e.  Grp )
34 simplrr 536 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  X )
3531, 32, 3syl2anc 411 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  z  e.  X )
36 eqid 2177 . . . . . . . . . . 11  |-  ( -g `  G )  =  (
-g `  G )
371, 10, 36grppncan 12893 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  z  e.  X )  ->  ( ( w  .+  z ) ( -g `  G ) z )  =  w )
3833, 34, 35, 37syl3anc 1238 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  =  w )
39 simpr 110 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( w  .+  z )  e.  S
)
4036subgsubcl 12976 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (
w  .+  z )  e.  S  /\  z  e.  S )  ->  (
( w  .+  z
) ( -g `  G
) z )  e.  S )
4131, 39, 32, 40syl3anc 1238 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( (
w  .+  z )
( -g `  G ) z )  e.  S
)
4238, 41eqeltrrd 2255 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  w  e.  S )
4310subgcl 12975 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S  /\  w  e.  S )  ->  (
z  .+  w )  e.  S )
4431, 32, 42, 43syl3anc 1238 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  /\  ( w  .+  z )  e.  S
)  ->  ( z  .+  w )  e.  S
)
4530, 44impbida 596 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  (
z  e.  S  /\  w  e.  X )
)  ->  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) )
4645anassrs 400 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  /\  w  e.  X )  ->  (
( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
) )
4746ralrimiva 2550 . . . 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 12999 . . . 4  |-  ( z  e.  N  <->  ( z  e.  X  /\  A. w  e.  X  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
) ) )
503, 47, 49sylanbrc 417 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  z  e.  S )  ->  z  e.  N )
5150ex 115 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  ( z  e.  S  ->  z  e.  N ) )
5251ssrdv 3161 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459    C_ wss 3129   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   0gc0g 12693   Grpcgrp 12809   invgcminusg 12810   -gcsg 12811  SubGrpcsubg 12958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-sbg 12814  df-subg 12961
This theorem is referenced by:  nmznsg  13004
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