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Theorem ghmnsgpreima 16105
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 16047 . . 3  |-  ( V  e.  (NrmSGrp `  T
)  ->  V  e.  (SubGrp `  T ) )
2 ghmpreima 16102 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
31, 2sylan2 474 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (NrmSGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
4 ghmgrp1 16083 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
54ad2antrr 725 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  S  e.  Grp )
6 simprl 755 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  x  e.  (
Base `  S )
)
7 simprr 756 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  ( `' F " V ) )
8 simpll 753 . . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
119, 10ghmf 16085 . . . . . . . . . . 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 5731 . . . . . . . . . 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 6002 . . . . . . . . 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 210 . . . . . . 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 459 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  y  e.  (
Base `  S )
)
19 eqid 2467 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
209, 19grpcl 15877 . . . . . 6  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +g  `  S
) y )  e.  ( Base `  S
) )
215, 6, 18, 20syl3anc 1228 . . . . 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 2467 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
239, 22grpsubcl 15932 . . . . 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 1228 . . . 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 2467 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
269, 22, 25ghmsub 16089 . . . . . . 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 1228 . . . . . 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 2467 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
299, 19, 28ghmlin 16086 . . . . . . . 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 1228 . . . . . . 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 6300 . . . . . 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 2508 . . . . 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 754 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  V  e.  (NrmSGrp `  T ) )
3412, 6ffvelrnd 6023 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (NrmSGrp `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( `' F " V ) ) )  ->  ( F `  x )  e.  (
Base `  T )
)
3517simprd 463 . . . . . 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 16048 . . . . . 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 1228 . . . . 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 2555 . . . 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 6002 . . . . 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 920 . . 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 2885 . 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 16049 . 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 664 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   `'ccnv 4998   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   Grpcgrp 15730   -gcsg 15733  SubGrpcsubg 16009  NrmSGrpcnsg 16010    GrpHom cghm 16078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-nsg 16013  df-ghm 16079
This theorem is referenced by:  ghmker  16106
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