Theorem conjnsg 17696

Description: A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
conjghm.x	|- X = ( Base ` G )
conjghm.p	|- .+ = ( +g ` G )
conjghm.m	|- .- = ( -g ` G )
conjsubg.f	|- F = ( x e. S |-> ( ( A .+ x ) .- A ) )

Assertion

Ref	Expression
conjnsg	|- ( ( S e. (NrmSGrp ` G ) /\ A e. X ) -> S = ran F )

Distinct variable groups:   x, .-   x, .+   x, A   x, G   x, S   x, X

Allowed substitution hint:   F( x)

Proof of Theorem conjnsg

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } = { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) }
2		conjghm.x	. . . . . 6 |- X = ( Base ` G )
3		conjghm.p	. . . . . 6 |- .+ = ( +g ` G )
4	1, 2, 3	isnsg4 17637	. . . . 5 |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } = X ) )
5	4	simprbi 480	. . . 4 |- ( S e. (NrmSGrp ` G ) -> { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } = X )
6	5	eleq2d 2687	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( A e. { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } <-> A e. X ) )
7	6	biimpar 502	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X ) -> A e. { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } )
8		nsgsubg 17626	. . 3 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
9		conjghm.m	. . . 4 |- .- = ( -g ` G )
10		conjsubg.f	. . . 4 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
11	2, 3, 9, 10, 1	conjnmz 17694	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } ) -> S = ran F )
12	8, 11	sylan 488	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } ) -> S = ran F )
13	7, 12	syldan 487	1 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X ) -> S = ran F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   -gcsg 17424  SubGrpcsubg 17588  NrmSGrpcnsg 17589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-nsg 17592

This theorem is referenced by:  sylow3lem6  18047