Theorem conjsubgen 17693

Description: A conjugated subgroup is equinumerous to the original subgroup. (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
conjsubgen	|- ( ( S e. (SubGrp ` 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 conjsubgen

Step	Hyp	Ref	Expression
1		subgrcl 17599	. . . . . . . 8 |- ( S e. (SubGrp ` G ) -> G e. Grp )
2		conjghm.x	. . . . . . . . 9 |- X = ( Base ` G )
3		conjghm.p	. . . . . . . . 9 |- .+ = ( +g ` G )
4		conjghm.m	. . . . . . . . 9 |- .- = ( -g ` G )
5		eqid 2622	. . . . . . . . 9 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
6	2, 3, 4, 5	conjghm 17691	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) e. ( G GrpHom G ) /\ ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-onto-> X ) )
7	1, 6	sylan 488	. . . . . . 7 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) e. ( G GrpHom G ) /\ ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-onto-> X ) )
8	7	simprd 479	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-onto-> X )
9		f1of1 6136	. . . . . 6 |- ( ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-onto-> X -> ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-> X )
10	8, 9	syl 17	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-> X )
11	2	subgss 17595	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S C_ X )
12	11	adantr 481	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S C_ X )
13		f1ssres 6108	. . . . 5 |- ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-> X /\ S C_ X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X )
14	10, 12, 13	syl2anc 693	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X )
15	12	resmptd 5452	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ( x e. S |-> ( ( A .+ x ) .- A ) ) )
16		conjsubg.f	. . . . . 6 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
17	15, 16	syl6eqr 2674	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
18		f1eq1 6096	. . . . 5 |- ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F -> ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X <-> F : S -1-1-> X ) )
19	17, 18	syl 17	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X <-> F : S -1-1-> X ) )
20	14, 19	mpbid 222	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> F : S -1-1-> X )
21		f1f1orn 6148	. . 3 |- ( F : S -1-1-> X -> F : S -1-1-onto-> ran F )
22	20, 21	syl 17	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> F : S -1-1-onto-> ran F )
23		f1oeng 7974	. 2 |- ( ( S e. (SubGrp ` G ) /\ F : S -1-1-onto-> ran F ) -> S ~~ ran F )
24	22, 23	syldan 487	1 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S ~~ ran F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588   GrpHom cghm 17657

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-en 7956  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-ghm 17658

This theorem is referenced by:  slwhash  18039  sylow2  18041  sylow3lem1  18042