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Theorem ghmnsgpreima 16858
Description: The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
ghmnsgpreima  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  ( `' F " V )  e.  (NrmSGrp `  S )
)

Proof of Theorem ghmnsgpreima
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgsubg 16800 . . 3  |-  ( V  e.  (NrmSGrp `  T
)  ->  V  e.  (SubGrp `  T ) )
2 ghmpreima 16855 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
31, 2sylan2 476 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
4 ghmgrp1 16836 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
54ad2antrr 730 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  S  e.  Grp )
6 simprl 762 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  x  e.  (
Base `  S )
)
7 simprr 764 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  ( `' F " V ) )
8 simpll 758 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  F  e.  ( S  GrpHom  T ) )
9 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
119, 10ghmf 16838 . . . . . . . . . . 11  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
128, 11syl 17 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  F : (
Base `  S ) --> ( Base `  T )
)
13 ffn 5746 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1412, 13syl 17 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  F  Fn  ( Base `  S ) )
15 elpreima 6017 . . . . . . . . 9  |-  ( F  Fn  ( Base `  S
)  ->  ( y  e.  ( `' F " V )  <->  ( y  e.  ( Base `  S
)  /\  ( F `  y )  e.  V
) ) )
1614, 15syl 17 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( y  e.  ( `' F " V )  <->  ( y  e.  ( Base `  S
)  /\  ( F `  y )  e.  V
) ) )
177, 16mpbid 213 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( y  e.  ( Base `  S
)  /\  ( F `  y )  e.  V
) )
1817simpld 460 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  (
Base `  S )
)
19 eqid 2429 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
209, 19grpcl 16630 . . . . . 6  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +g  `  S
) y )  e.  ( Base `  S
) )
215, 6, 18, 20syl3anc 1264 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( x ( +g  `  S ) y )  e.  (
Base `  S )
)
22 eqid 2429 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
239, 22grpsubcl 16685 . . . . 5  |-  ( ( S  e.  Grp  /\  ( x ( +g  `  S ) y )  e.  ( Base `  S
)  /\  x  e.  ( Base `  S )
)  ->  ( (
x ( +g  `  S
) y ) (
-g `  S )
x )  e.  (
Base `  S )
)
245, 21, 6, 23syl3anc 1264 . . . 4  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( ( x ( +g  `  S
) y ) (
-g `  S )
x )  e.  (
Base `  S )
)
25 eqid 2429 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
269, 22, 25ghmsub 16842 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  (
x ( +g  `  S
) y )  e.  ( Base `  S
)  /\  x  e.  ( Base `  S )
)  ->  ( F `  ( ( x ( +g  `  S ) y ) ( -g `  S ) x ) )  =  ( ( F `  ( x ( +g  `  S
) y ) ) ( -g `  T
) ( F `  x ) ) )
278, 21, 6, 26syl3anc 1264 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  ( ( x ( +g  `  S ) y ) ( -g `  S ) x ) )  =  ( ( F `  ( x ( +g  `  S
) y ) ) ( -g `  T
) ( F `  x ) ) )
28 eqid 2429 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
299, 19, 28ghmlin 16839 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
308, 6, 18, 29syl3anc 1264 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
3130oveq1d 6320 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( ( F `
 ( x ( +g  `  S ) y ) ) (
-g `  T )
( F `  x
) )  =  ( ( ( F `  x ) ( +g  `  T ) ( F `
 y ) ) ( -g `  T
) ( F `  x ) ) )
3227, 31eqtrd 2470 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  ( ( x ( +g  `  S ) y ) ( -g `  S ) x ) )  =  ( ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) (
-g `  T )
( F `  x
) ) )
33 simplr 760 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  V  e.  (NrmSGrp `  T ) )
3412, 6ffvelrnd 6038 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  x )  e.  (
Base `  T )
)
3517simprd 464 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  y )  e.  V
)
3610, 28, 25nsgconj 16801 . . . . . 6  |-  ( ( V  e.  (NrmSGrp `  T
)  /\  ( F `  x )  e.  (
Base `  T )  /\  ( F `  y
)  e.  V )  ->  ( ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) (
-g `  T )
( F `  x
) )  e.  V
)
3733, 34, 35, 36syl3anc 1264 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) (
-g `  T )
( F `  x
) )  e.  V
)
3832, 37eqeltrd 2517 . . . 4  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  ( ( x ( +g  `  S ) y ) ( -g `  S ) x ) )  e.  V )
39 elpreima 6017 . . . . 5  |-  ( F  Fn  ( Base `  S
)  ->  ( (
( x ( +g  `  S ) y ) ( -g `  S
) x )  e.  ( `' F " V )  <->  ( (
( x ( +g  `  S ) y ) ( -g `  S
) x )  e.  ( Base `  S
)  /\  ( F `  ( ( x ( +g  `  S ) y ) ( -g `  S ) x ) )  e.  V ) ) )
4014, 39syl 17 . . . 4  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( ( ( x ( +g  `  S
) y ) (
-g `  S )
x )  e.  ( `' F " V )  <-> 
( ( ( x ( +g  `  S
) y ) (
-g `  S )
x )  e.  (
Base `  S )  /\  ( F `  (
( x ( +g  `  S ) y ) ( -g `  S
) x ) )  e.  V ) ) )
4124, 38, 40mpbir2and 930 . . 3  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( ( x ( +g  `  S
) y ) (
-g `  S )
x )  e.  ( `' F " V ) )
4241ralrimivva 2853 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  A. x  e.  ( Base `  S
) A. y  e.  ( `' F " V ) ( ( x ( +g  `  S
) y ) (
-g `  S )
x )  e.  ( `' F " V ) )
439, 19, 22isnsg3 16802 . 2  |-  ( ( `' F " V )  e.  (NrmSGrp `  S
)  <->  ( ( `' F " V )  e.  (SubGrp `  S
)  /\  A. x  e.  ( Base `  S
) A. y  e.  ( `' F " V ) ( ( x ( +g  `  S
) y ) (
-g `  S )
x )  e.  ( `' F " V ) ) )
443, 42, 43sylanbrc 668 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  ( `' F " V )  e.  (NrmSGrp `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   `'ccnv 4853   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   Grpcgrp 16620   -gcsg 16622  SubGrpcsubg 16762  NrmSGrpcnsg 16763    GrpHom cghm 16831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-nsg 16766  df-ghm 16832
This theorem is referenced by:  ghmker  16859
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