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Theorem conjsubg 15799
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
StepHypRef Expression
1 conjghm.x . . . . 5  |-  X  =  ( Base `  G
)
21subgss 15703 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
32adantr 465 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  C_  X )
4 df-ima 4874 . . . 4  |-  ( ( x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) " S )  =  ran  ( ( x  e.  X  |->  ( ( A  .+  x
)  .-  A )
)  |`  S )
5 resmpt 5177 . . . . . 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
) )
75, 6syl6eqr 2493 . . . . 5  |-  ( S 
C_  X  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F )
87rneqd 5088 . . . 4  |-  ( S 
C_  X  ->  ran  ( ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )  |`  S )  =  ran  F )
94, 8syl5eq 2487 . . 3  |-  ( S 
C_  X  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) " S
)  =  ran  F
)
103, 9syl 16 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) " S
)  =  ran  F
)
11 subgrcl 15707 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
12 conjghm.p . . . . . 6  |-  .+  =  ( +g  `  G )
13 conjghm.m . . . . . 6  |-  .-  =  ( -g `  G )
14 eqid 2443 . . . . . 6  |-  ( x  e.  X  |->  ( ( A  .+  x ) 
.-  A ) )  =  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )
151, 12, 13, 14conjghm 15798 . . . . 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 ) )
1611, 15sylan 471 . . . 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
) )
1716simpld 459 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
)  e.  ( G 
GrpHom  G ) )
18 simpl 457 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  e.  (SubGrp `  G )
)
19 ghmima 15788 . . 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 ) )
2017, 18, 19syl2anc 661 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) " S
)  e.  (SubGrp `  G ) )
2110, 20eqeltrrd 2518 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  ran  F  e.  (SubGrp `  G
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3349    e. cmpt 4371   ran crn 4862    |` cres 4863   "cima 4864   -1-1-onto->wf1o 5438   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   Grpcgrp 15431   -gcsg 15434  SubGrpcsubg 15696    GrpHom cghm 15765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-ghm 15766
This theorem is referenced by:  slwhash  16144  sylow2  16146  sylow3lem1  16147
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