Theorem conjsubg 13927

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 13824	. . . 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 4744	. . . 4 |- ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) = ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S )
5		resmpt 5067	. . . . . 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 2282	. . . . 5 |- ( S C_ X -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
8	7	rneqd 4967	. . . 4 |- ( S C_ X -> ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ran F )
9	4, 8	eqtrid 2276	. . 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 13829	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
12		conjghm.p	. . . . . 6 |- .+ = ( +g ` G )
13		conjghm.m	. . . . . 6 |- .- = ( -g ` G )
14		eqid 2231	. . . . . 6 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
15	1, 12, 13, 14	conjghm 13926	. . . . 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 13915	. . 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 2309	1 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ran F e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   C_ wss 3201   |-> cmpt 4155   ran crn 4732   |` cres 4733   "cima 4734   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   Grpcgrp 13646   -gcsg 13648  SubGrpcsubg 13817   GrpHom cghm 13890

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-sbg 13651  df-subg 13820  df-ghm 13891

This theorem is referenced by: (None)