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Theorem ghmnsgpreima 14723
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 14665 . . 3  |-  ( V  e.  (NrmSGrp `  T
)  ->  V  e.  (SubGrp `  T ) )
2 ghmpreima 14720 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
31, 2sylan2 460 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
4 ghmgrp1 14701 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
54ad2antrr 706 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  S  e.  Grp )
6 simprl 732 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  x  e.  (
Base `  S )
)
7 simprr 733 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  ( `' F " V ) )
8 simpll 730 . . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2296 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
119, 10ghmf 14703 . . . . . . . . . . 11  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
128, 11syl 15 . . . . . . . . . 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 5405 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1412, 13syl 15 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  F  Fn  ( Base `  S ) )
15 elpreima 5661 . . . . . . . . 9  |-  ( F  Fn  ( Base `  S
)  ->  ( y  e.  ( `' F " V )  <->  ( y  e.  ( Base `  S
)  /\  ( F `  y )  e.  V
) ) )
1614, 15syl 15 . . . . . . . 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 201 . . . . . . 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 445 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  (
Base `  S )
)
19 eqid 2296 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
209, 19grpcl 14511 . . . . . 6  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +g  `  S
) y )  e.  ( Base `  S
) )
215, 6, 18, 20syl3anc 1182 . . . . 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 2296 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
239, 22grpsubcl 14562 . . . . 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 1182 . . . 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 2296 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
269, 22, 25ghmsub 14707 . . . . . . 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 1182 . . . . . 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 2296 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
299, 19, 28ghmlin 14704 . . . . . . . 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 1182 . . . . . . 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 5889 . . . . . 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 2328 . . . . 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 731 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  V  e.  (NrmSGrp `  T ) )
34 ffvelrn 5679 . . . . . . 7  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  x  e.  ( Base `  S
) )  ->  ( F `  x )  e.  ( Base `  T
) )
3512, 6, 34syl2anc 642 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  x )  e.  (
Base `  T )
)
3617simprd 449 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  y )  e.  V
)
3710, 28, 25nsgconj 14666 . . . . . 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
)
3833, 35, 36, 37syl3anc 1182 . . . . 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
)
3932, 38eqeltrd 2370 . . . 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 )
40 elpreima 5661 . . . . 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 ) ) )
4114, 40syl 15 . . . 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 ) ) )
4224, 39, 41mpbir2and 888 . . 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 ) )
4342ralrimivva 2648 . 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 ) )
449, 19, 22isnsg3 14667 . 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 ) ) )
453, 43, 44sylanbrc 645 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631  NrmSGrpcnsg 14632    GrpHom cghm 14696
This theorem is referenced by:  ghmker  14724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635  df-ghm 14697
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