Theorem conjsubg 13728

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 13625	. . . 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 4706	. . . 4 |- ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) = ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S )
5		resmpt 5026	. . . . . 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 2258	. . . . 5 |- ( S C_ X -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
8	7	rneqd 4926	. . . 4 |- ( S C_ X -> ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ran F )
9	4, 8	eqtrid 2252	. . 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 13630	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
12		conjghm.p	. . . . . 6 |- .+ = ( +g ` G )
13		conjghm.m	. . . . . 6 |- .- = ( -g ` G )
14		eqid 2207	. . . . . 6 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
15	1, 12, 13, 14	conjghm 13727	. . . . 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 13716	. . 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 2285	1 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ran F e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   C_ wss 3174   |-> cmpt 4121   ran crn 4694   |` cres 4695   "cima 4696   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   Grpcgrp 13447   -gcsg 13449  SubGrpcsubg 13618   GrpHom cghm 13691

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-sbg 13452  df-subg 13621  df-ghm 13692

This theorem is referenced by: (None)