Theorem conjsubg 13584

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 13481	. . . 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 4687	. . . 4 |- ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) = ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S )
5		resmpt 5006	. . . . . 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 2255	. . . . 5 |- ( S C_ X -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
8	7	rneqd 4906	. . . 4 |- ( S C_ X -> ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ran F )
9	4, 8	eqtrid 2249	. . 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 13486	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
12		conjghm.p	. . . . . 6 |- .+ = ( +g ` G )
13		conjghm.m	. . . . . 6 |- .- = ( -g ` G )
14		eqid 2204	. . . . . 6 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
15	1, 12, 13, 14	conjghm 13583	. . . . 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 13572	. . 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 2282	1 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ran F e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   C_ wss 3165   |-> cmpt 4104   ran crn 4675   |` cres 4676   "cima 4677   -1-1-onto->wf1o 5269   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   Grpcgrp 13303   -gcsg 13305  SubGrpcsubg 13474   GrpHom cghm 13547

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-iress 12811  df-plusg 12893  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-sbg 13308  df-subg 13477  df-ghm 13548

This theorem is referenced by: (None)