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Theorem ghmnsgpreima 15022
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 14964 . . 3  |-  ( V  e.  (NrmSGrp `  T
)  ->  V  e.  (SubGrp `  T ) )
2 ghmpreima 15019 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
31, 2sylan2 461 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
4 ghmgrp1 15000 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
54ad2antrr 707 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  S  e.  Grp )
6 simprl 733 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  x  e.  (
Base `  S )
)
7 simprr 734 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  ( `' F " V ) )
8 simpll 731 . . . . . . . . . . 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 2435 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2435 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
119, 10ghmf 15002 . . . . . . . . . . 11  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
128, 11syl 16 . . . . . . . . . 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 5583 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1412, 13syl 16 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  F  Fn  ( Base `  S ) )
15 elpreima 5842 . . . . . . . . 9  |-  ( F  Fn  ( Base `  S
)  ->  ( y  e.  ( `' F " V )  <->  ( y  e.  ( Base `  S
)  /\  ( F `  y )  e.  V
) ) )
1614, 15syl 16 . . . . . . . 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 202 . . . . . . 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 446 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  (
Base `  S )
)
19 eqid 2435 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
209, 19grpcl 14810 . . . . . 6  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +g  `  S
) y )  e.  ( Base `  S
) )
215, 6, 18, 20syl3anc 1184 . . . . 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 2435 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
239, 22grpsubcl 14861 . . . . 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 1184 . . . 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 2435 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
269, 22, 25ghmsub 15006 . . . . . . 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 1184 . . . . . 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 2435 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
299, 19, 28ghmlin 15003 . . . . . . . 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 1184 . . . . . . 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 6088 . . . . . 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 2467 . . . . 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 732 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  V  e.  (NrmSGrp `  T ) )
3412, 6ffvelrnd 5863 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  x )  e.  (
Base `  T )
)
3517simprd 450 . . . . . 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 14965 . . . . . 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 1184 . . . . 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 2509 . . . 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 5842 . . . . 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 16 . . . 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 889 . . 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 2790 . 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 14966 . 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 646 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  ( `' F " V )  e.  (NrmSGrp `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   -gcsg 14680  SubGrpcsubg 14930  NrmSGrpcnsg 14931    GrpHom cghm 14995
This theorem is referenced by:  ghmker  15023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-nsg 14934  df-ghm 14996
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