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Theorem ghmnsgima 16616
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 999 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  F  e.  ( S  GrpHom  T ) )
2 nsgsubg 16559 . . . 4  |-  ( U  e.  (NrmSGrp `  S
)  ->  U  e.  (SubGrp `  S ) )
323ad2ant2 1021 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  U  e.  (SubGrp `  S ) )
4 ghmima 16613 . . 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 16595 . . . . . . . . 9  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
86, 7syl 17 . . . . . . . 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 758 . . . . . . . 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 2404 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
1110subgss 16528 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  ( Base `  S ) )
123, 11syl 17 . . . . . . . . . 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 760 . . . . . . . . 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 3445 . . . . . . . 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 2404 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
1710, 16grpcl 16389 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  z  e.  ( Base `  S )  /\  x  e.  ( Base `  S
) )  ->  (
z ( +g  `  S
) x )  e.  ( Base `  S
) )
188, 9, 15, 17syl3anc 1232 . . . . . . 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 2404 . . . . . . . 8  |-  ( -g `  S )  =  (
-g `  S )
20 eqid 2404 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
2110, 19, 20ghmsub 16601 . . . . . . 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 1232 . . . . . 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 2404 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
2410, 16, 23ghmlin 16598 . . . . . . . 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 1232 . . . . . . 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 6295 . . . . . 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 2445 . . . . 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 16597 . . . . . . . . 9  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> Y )
301, 29syl 17 . . . . . . . 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 5716 . . . . . . 7  |-  ( F : ( Base `  S
) --> Y  ->  F  Fn  ( Base `  S
) )
3331, 32syl 17 . . . . . 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 1003 . . . . . . 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 16560 . . . . . . 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 1232 . . . . . 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 6133 . . . . . 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 1232 . . . . 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 2493 . . . 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 2827 . . 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 17 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  F  Fn  ( Base `  S )
)
42 oveq1 6287 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  T
) y )  =  ( ( F `  z ) ( +g  `  T ) y ) )
43 id 23 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  x  =  ( F `  z ) )
4442, 43oveq12d 6298 . . . . . . . 8  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  T ) y ) ( -g `  T
) x )  =  ( ( ( F `
 z ) ( +g  `  T ) y ) ( -g `  T ) ( F `
 z ) ) )
4544eleq1d 2473 . . . . . . 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 2845 . . . . . 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 6014 . . . . 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 17 . . . 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 1001 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ran  F  =  Y )
5049raleqdv 3012 . . . 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 6288 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  T
) y )  =  ( ( F `  z ) ( +g  `  T ) ( F `
 x ) ) )
5251oveq1d 6295 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  T ) y ) ( -g `  T
) ( F `  z ) )  =  ( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) ) )
5352eleq1d 2473 . . . . . . 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 6135 . . . . . 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 2845 . . . 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 285 . . 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 234 . 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 16561 . 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756    C_ wss 3416   ran crn 4826   "cima 4828    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   Grpcgrp 16379   -gcsg 16381  SubGrpcsubg 16521  NrmSGrpcnsg 16522    GrpHom cghm 16590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-grp 16383  df-minusg 16384  df-sbg 16385  df-subg 16524  df-nsg 16525  df-ghm 16591
This theorem is referenced by: (None)
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