Theorem conjsubg 13613

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 13510	. . . 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 4688	. . . 4 |- ( ( x e. X |-> ( ( A .+ x ) .- A ) ) " S ) = ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S )
5		resmpt 5007	. . . . . 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 2256	. . . . 5 |- ( S C_ X -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
8	7	rneqd 4907	. . . 4 |- ( S C_ X -> ran ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ran F )
9	4, 8	eqtrid 2250	. . 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 13515	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
12		conjghm.p	. . . . . 6 |- .+ = ( +g ` G )
13		conjghm.m	. . . . . 6 |- .- = ( -g ` G )
14		eqid 2205	. . . . . 6 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
15	1, 12, 13, 14	conjghm 13612	. . . . 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 13601	. . 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 2283	1 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ran F e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   C_ wss 3166   |-> cmpt 4105   ran crn 4676   |` cres 4677   "cima 4678   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   Grpcgrp 13332   -gcsg 13334  SubGrpcsubg 13503   GrpHom cghm 13576

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-sbg 13337  df-subg 13506  df-ghm 13577

This theorem is referenced by: (None)