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Theorem conjsubg 16497
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 16401 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
32adantr 463 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  C_  X )
4 df-ima 5001 . . . 4  |-  ( ( x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) " S )  =  ran  ( ( x  e.  X  |->  ( ( A  .+  x
)  .-  A )
)  |`  S )
5 resmpt 5311 . . . . . 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 2513 . . . . 5  |-  ( S 
C_  X  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F )
87rneqd 5219 . . . 4  |-  ( S 
C_  X  ->  ran  ( ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )  |`  S )  =  ran  F )
94, 8syl5eq 2507 . . 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 16405 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
12 conjghm.p . . . . . 6  |-  .+  =  ( +g  `  G )
13 conjghm.m . . . . . 6  |-  .-  =  ( -g `  G )
14 eqid 2454 . . . . . 6  |-  ( x  e.  X  |->  ( ( A  .+  x ) 
.-  A ) )  =  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )
151, 12, 13, 14conjghm 16496 . . . . 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 469 . . . 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 457 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
)  e.  ( G 
GrpHom  G ) )
18 simpl 455 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  e.  (SubGrp `  G )
)
19 ghmima 16486 . . 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 659 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) " S
)  e.  (SubGrp `  G ) )
2110, 20eqeltrrd 2543 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  ran  F  e.  (SubGrp `  G
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461    |-> cmpt 4497   ran crn 4989    |` cres 4990   "cima 4991   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   Grpcgrp 16252   -gcsg 16254  SubGrpcsubg 16394    GrpHom cghm 16463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-ghm 16464
This theorem is referenced by:  slwhash  16843  sylow2  16845  sylow3lem1  16846
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