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Theorem ghmnsgpreima 14957
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 14899 . . 3  |-  ( V  e.  (NrmSGrp `  T
)  ->  V  e.  (SubGrp `  T ) )
2 ghmpreima 14954 . . 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 14935 . . . . . 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 2387 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2387 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
119, 10ghmf 14937 . . . . . . . . . . 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 5531 . . . . . . . . . 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 5789 . . . . . . . . 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 2387 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
209, 19grpcl 14745 . . . . . 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 2387 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
239, 22grpsubcl 14796 . . . . 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 2387 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
269, 22, 25ghmsub 14941 . . . . . . 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 2387 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
299, 19, 28ghmlin 14938 . . . . . . . 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 6035 . . . . . 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 2419 . . . . 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 5810 . . . . . 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 14900 . . . . . 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 2461 . . . 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 5789 . . . . 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 2741 . 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 14901 . 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 1649    e. wcel 1717   A.wral 2649   `'ccnv 4817   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   Grpcgrp 14612   -gcsg 14615  SubGrpcsubg 14865  NrmSGrpcnsg 14866    GrpHom cghm 14930
This theorem is referenced by:  ghmker  14958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-nsg 14869  df-ghm 14931
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