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Theorem ghmnsgima 16078
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 991 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  F  e.  ( S  GrpHom  T ) )
2 nsgsubg 16021 . . . 4  |-  ( U  e.  (NrmSGrp `  S
)  ->  U  e.  (SubGrp `  S ) )
323ad2ant2 1013 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  U  e.  (SubGrp `  S ) )
4 ghmima 16075 . . 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 16057 . . . . . . . . 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 2460 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
1110subgss 15990 . . . . . . . . . . 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 3498 . . . . . . . 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 2460 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
1710, 16grpcl 15857 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  z  e.  ( Base `  S )  /\  x  e.  ( Base `  S
) )  ->  (
z ( +g  `  S
) x )  e.  ( Base `  S
) )
188, 9, 15, 17syl3anc 1223 . . . . . . 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 2460 . . . . . . . 8  |-  ( -g `  S )  =  (
-g `  S )
20 eqid 2460 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
2110, 19, 20ghmsub 16063 . . . . . . 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 1223 . . . . . 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 2460 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
2410, 16, 23ghmlin 16060 . . . . . . . 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 1223 . . . . . . 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 6290 . . . . . 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 2501 . . . . 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 16059 . . . . . . . . 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 5722 . . . . . . 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 995 . . . . . . 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 16022 . . . . . . 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 1223 . . . . . 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 6129 . . . . . 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 1223 . . . . 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 2549 . . . 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 2878 . . 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 6282 . . . . . . . . 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 6293 . . . . . . . 8  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  T ) y ) ( -g `  T
) x )  =  ( ( ( F `
 z ) ( +g  `  T ) y ) ( -g `  T ) ( F `
 z ) ) )
4544eleq1d 2529 . . . . . . 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 2896 . . . . . 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 6015 . . . . 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 993 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ran  F  =  Y )
5049raleqdv 3057 . . . 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 6283 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  T
) y )  =  ( ( F `  z ) ( +g  `  T ) ( F `
 x ) ) )
5251oveq1d 6290 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  T ) y ) ( -g `  T
) ( F `  z ) )  =  ( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) ) )
5352eleq1d 2529 . . . . . . 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 6131 . . . . . 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 2896 . . . 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 16023 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   ran crn 4993   "cima 4995    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716   -gcsg 15719  SubGrpcsubg 15983  NrmSGrpcnsg 15984    GrpHom cghm 16052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-nsg 15987  df-ghm 16053
This theorem is referenced by: (None)
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