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Theorem ghmnsgima 15761
Description: The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypothesis
Ref Expression
ghmnsgima.1  |-  Y  =  ( Base `  T
)
Assertion
Ref Expression
ghmnsgima  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( F " U )  e.  (NrmSGrp `  T ) )

Proof of Theorem ghmnsgima
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  F  e.  ( S  GrpHom  T ) )
2 nsgsubg 15704 . . . 4  |-  ( U  e.  (NrmSGrp `  S
)  ->  U  e.  (SubGrp `  S ) )
323ad2ant2 1010 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  U  e.  (SubGrp `  S ) )
4 ghmima 15758 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (SubGrp `  S )
)  ->  ( F " U )  e.  (SubGrp `  T ) )
51, 3, 4syl2anc 661 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( F " U )  e.  (SubGrp `  T ) )
61adantr 465 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  F  e.  ( S  GrpHom  T ) )
7 ghmgrp1 15740 . . . . . . . . 9  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
86, 7syl 16 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  S  e.  Grp )
9 simprl 755 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
z  e.  ( Base `  S ) )
10 eqid 2438 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
1110subgss 15673 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  ( Base `  S ) )
123, 11syl 16 . . . . . . . . . 10  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  U  C_  ( Base `  S ) )
1312adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  U  C_  ( Base `  S
) )
14 simprr 756 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  x  e.  U )
1513, 14sseldd 3352 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  x  e.  ( Base `  S ) )
16 eqid 2438 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
1710, 16grpcl 15542 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  z  e.  ( Base `  S )  /\  x  e.  ( Base `  S
) )  ->  (
z ( +g  `  S
) x )  e.  ( Base `  S
) )
188, 9, 15, 17syl3anc 1218 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( z ( +g  `  S ) x )  e.  ( Base `  S
) )
19 eqid 2438 . . . . . . . 8  |-  ( -g `  S )  =  (
-g `  S )
20 eqid 2438 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
2110, 19, 20ghmsub 15746 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  (
z ( +g  `  S
) x )  e.  ( Base `  S
)  /\  z  e.  ( Base `  S )
)  ->  ( F `  ( ( z ( +g  `  S ) x ) ( -g `  S ) z ) )  =  ( ( F `  ( z ( +g  `  S
) x ) ) ( -g `  T
) ( F `  z ) ) )
226, 18, 9, 21syl3anc 1218 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( F `  (
( z ( +g  `  S ) x ) ( -g `  S
) z ) )  =  ( ( F `
 ( z ( +g  `  S ) x ) ) (
-g `  T )
( F `  z
) ) )
23 eqid 2438 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
2410, 16, 23ghmlin 15743 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  z  e.  ( Base `  S
)  /\  x  e.  ( Base `  S )
)  ->  ( F `  ( z ( +g  `  S ) x ) )  =  ( ( F `  z ) ( +g  `  T
) ( F `  x ) ) )
256, 9, 15, 24syl3anc 1218 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( F `  (
z ( +g  `  S
) x ) )  =  ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) )
2625oveq1d 6101 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( ( F `  ( z ( +g  `  S ) x ) ) ( -g `  T
) ( F `  z ) )  =  ( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) ) )
2722, 26eqtrd 2470 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( F `  (
( z ( +g  `  S ) x ) ( -g `  S
) z ) )  =  ( ( ( F `  z ) ( +g  `  T
) ( F `  x ) ) (
-g `  T )
( F `  z
) ) )
28 ghmnsgima.1 . . . . . . . . . 10  |-  Y  =  ( Base `  T
)
2910, 28ghmf 15742 . . . . . . . . 9  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> Y )
301, 29syl 16 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  F :
( Base `  S ) --> Y )
3130adantr 465 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  F : ( Base `  S
) --> Y )
32 ffn 5554 . . . . . . 7  |-  ( F : ( Base `  S
) --> Y  ->  F  Fn  ( Base `  S
) )
3331, 32syl 16 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  F  Fn  ( Base `  S ) )
34 simpl2 992 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  ->  U  e.  (NrmSGrp `  S
) )
3510, 16, 19nsgconj 15705 . . . . . . 7  |-  ( ( U  e.  (NrmSGrp `  S
)  /\  z  e.  ( Base `  S )  /\  x  e.  U
)  ->  ( (
z ( +g  `  S
) x ) (
-g `  S )
z )  e.  U
)
3634, 9, 14, 35syl3anc 1218 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( ( z ( +g  `  S ) x ) ( -g `  S ) z )  e.  U )
37 fnfvima 5950 . . . . . 6  |-  ( ( F  Fn  ( Base `  S )  /\  U  C_  ( Base `  S
)  /\  ( (
z ( +g  `  S
) x ) (
-g `  S )
z )  e.  U
)  ->  ( F `  ( ( z ( +g  `  S ) x ) ( -g `  S ) z ) )  e.  ( F
" U ) )
3833, 13, 36, 37syl3anc 1218 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( F `  (
( z ( +g  `  S ) x ) ( -g `  S
) z ) )  e.  ( F " U ) )
3927, 38eqeltrrd 2513 . . . 4  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  U  e.  (NrmSGrp `  S
)  /\  ran  F  =  Y )  /\  (
z  e.  ( Base `  S )  /\  x  e.  U ) )  -> 
( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) )  e.  ( F " U ) )
4039ralrimivva 2803 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  A. z  e.  ( Base `  S
) A. x  e.  U  ( ( ( F `  z ) ( +g  `  T
) ( F `  x ) ) (
-g `  T )
( F `  z
) )  e.  ( F " U ) )
4130, 32syl 16 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  F  Fn  ( Base `  S )
)
42 oveq1 6093 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  T
) y )  =  ( ( F `  z ) ( +g  `  T ) y ) )
43 id 22 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  x  =  ( F `  z ) )
4442, 43oveq12d 6104 . . . . . . . 8  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  T ) y ) ( -g `  T
) x )  =  ( ( ( F `
 z ) ( +g  `  T ) y ) ( -g `  T ) ( F `
 z ) ) )
4544eleq1d 2504 . . . . . . 7  |-  ( x  =  ( F `  z )  ->  (
( ( x ( +g  `  T ) y ) ( -g `  T ) x )  e.  ( F " U )  <->  ( (
( F `  z
) ( +g  `  T
) y ) (
-g `  T )
( F `  z
) )  e.  ( F " U ) ) )
4645ralbidv 2730 . . . . . 6  |-  ( x  =  ( F `  z )  ->  ( A. y  e.  ( F " U ) ( ( x ( +g  `  T ) y ) ( -g `  T
) x )  e.  ( F " U
)  <->  A. y  e.  ( F " U ) ( ( ( F `
 z ) ( +g  `  T ) y ) ( -g `  T ) ( F `
 z ) )  e.  ( F " U ) ) )
4746ralrn 5841 . . . . 5  |-  ( F  Fn  ( Base `  S
)  ->  ( A. x  e.  ran  F A. y  e.  ( F " U ) ( ( x ( +g  `  T
) y ) (
-g `  T )
x )  e.  ( F " U )  <->  A. z  e.  ( Base `  S ) A. y  e.  ( F " U ) ( ( ( F `  z
) ( +g  `  T
) y ) (
-g `  T )
( F `  z
) )  e.  ( F " U ) ) )
4841, 47syl 16 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( A. x  e.  ran  F A. y  e.  ( F " U ) ( ( x ( +g  `  T
) y ) (
-g `  T )
x )  e.  ( F " U )  <->  A. z  e.  ( Base `  S ) A. y  e.  ( F " U ) ( ( ( F `  z
) ( +g  `  T
) y ) (
-g `  T )
( F `  z
) )  e.  ( F " U ) ) )
49 simp3 990 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ran  F  =  Y )
5049raleqdv 2918 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( A. x  e.  ran  F A. y  e.  ( F " U ) ( ( x ( +g  `  T
) y ) (
-g `  T )
x )  e.  ( F " U )  <->  A. x  e.  Y  A. y  e.  ( F " U ) ( ( x ( +g  `  T ) y ) ( -g `  T
) x )  e.  ( F " U
) ) )
51 oveq2 6094 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  T
) y )  =  ( ( F `  z ) ( +g  `  T ) ( F `
 x ) ) )
5251oveq1d 6101 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  T ) y ) ( -g `  T
) ( F `  z ) )  =  ( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) ) )
5352eleq1d 2504 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
( ( ( F `
 z ) ( +g  `  T ) y ) ( -g `  T ) ( F `
 z ) )  e.  ( F " U )  <->  ( (
( F `  z
) ( +g  `  T
) ( F `  x ) ) (
-g `  T )
( F `  z
) )  e.  ( F " U ) ) )
5453ralima 5952 . . . . . 6  |-  ( ( F  Fn  ( Base `  S )  /\  U  C_  ( Base `  S
) )  ->  ( A. y  e.  ( F " U ) ( ( ( F `  z ) ( +g  `  T ) y ) ( -g `  T
) ( F `  z ) )  e.  ( F " U
)  <->  A. x  e.  U  ( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) )  e.  ( F " U ) ) )
5541, 12, 54syl2anc 661 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( A. y  e.  ( F " U ) ( ( ( F `  z
) ( +g  `  T
) y ) (
-g `  T )
( F `  z
) )  e.  ( F " U )  <->  A. x  e.  U  ( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) )  e.  ( F " U ) ) )
5655ralbidv 2730 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( A. z  e.  ( Base `  S ) A. y  e.  ( F " U
) ( ( ( F `  z ) ( +g  `  T
) y ) (
-g `  T )
( F `  z
) )  e.  ( F " U )  <->  A. z  e.  ( Base `  S ) A. x  e.  U  (
( ( F `  z ) ( +g  `  T ) ( F `
 x ) ) ( -g `  T
) ( F `  z ) )  e.  ( F " U
) ) )
5748, 50, 563bitr3d 283 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( A. x  e.  Y  A. y  e.  ( F " U ) ( ( x ( +g  `  T
) y ) (
-g `  T )
x )  e.  ( F " U )  <->  A. z  e.  ( Base `  S ) A. x  e.  U  (
( ( F `  z ) ( +g  `  T ) ( F `
 x ) ) ( -g `  T
) ( F `  z ) )  e.  ( F " U
) ) )
5840, 57mpbird 232 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  A. x  e.  Y  A. y  e.  ( F " U
) ( ( x ( +g  `  T
) y ) (
-g `  T )
x )  e.  ( F " U ) )
5928, 23, 20isnsg3 15706 . 2  |-  ( ( F " U )  e.  (NrmSGrp `  T
)  <->  ( ( F
" U )  e.  (SubGrp `  T )  /\  A. x  e.  Y  A. y  e.  ( F " U ) ( ( x ( +g  `  T ) y ) ( -g `  T
) x )  e.  ( F " U
) ) )
605, 58, 59sylanbrc 664 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( F " U )  e.  (NrmSGrp `  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710    C_ wss 3323   ran crn 4836   "cima 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   Grpcgrp 15402   -gcsg 15405  SubGrpcsubg 15666  NrmSGrpcnsg 15667    GrpHom cghm 15735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-nsg 15670  df-ghm 15736
This theorem is referenced by: (None)
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