Theorem conjsubg 13347

Description: A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-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
conjsubg	|- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ran F e. (SubGrp ` G ) )

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

Allowed substitution hint:   F( x)

Proof of Theorem conjsubg

Step	Hyp	Ref	Expression
1		conjghm.x	. . . . 5 |- X = ( Base ` G )
2	1	subgss 13244	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ X )
3	2	adantr 276	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S C_ X )
4		df-ima 4672	. . . 4 |- ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) = ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S )
5		resmpt 4990	. . . . . 6 |- ( S C_ X -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ( x e. S |-> ( ( A .+ x ) .- A ) ) )
6		conjsubg.f	. . . . . 6 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
7	5, 6	eqtr4di 2244	. . . . 5 |- ( S C_ X -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
8	7	rneqd 4891	. . . 4 |- ( S C_ X -> ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ran F )
9	4, 8	eqtrid 2238	. . 3 |- ( S C_ X -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) = ran F )
10	3, 9	syl 14	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) = ran F )
11		subgrcl 13249	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
12		conjghm.p	. . . . . 6 |- .+ = ( +g ` G )
13		conjghm.m	. . . . . 6 |- .- = ( -g ` G )
14		eqid 2193	. . . . . 6 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
15	1, 12, 13, 14	conjghm 13346	. . . . 5 |- ( ( 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 ) )
16	11, 15	sylan 283	. . . 4 |- ( ( 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 ) )
17	16	simpld 112	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( x e. X |-> ( ( A .+ x ) .- A ) ) e. ( G GrpHom G ) )
18		simpl 109	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S e. (SubGrp ` G ) )
19		ghmima 13335	. . 3 |- ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) e. ( G GrpHom G ) /\ S e. (SubGrp ` G ) ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) e. (SubGrp ` G ) )
20	17, 18, 19	syl2anc 411	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) e. (SubGrp ` G ) )
21	10, 20	eqeltrrd 2271	1 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ran F e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3153   |-> cmpt 4090   ran crn 4660   |` cres 4661   "cima 4662   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   -gcsg 13074  SubGrpcsubg 13237   GrpHom cghm 13310

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-subg 13240  df-ghm 13311

This theorem is referenced by: (None)