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Theorem ghmnsgima 15890
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 15833 . . . 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 15887 . . 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 15869 . . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
1110subgss 15802 . . . . . . . . . . 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 3466 . . . . . . . 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 2454 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
1710, 16grpcl 15671 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  z  e.  ( Base `  S )  /\  x  e.  ( Base `  S
) )  ->  (
z ( +g  `  S
) x )  e.  ( Base `  S
) )
188, 9, 15, 17syl3anc 1219 . . . . . . 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 2454 . . . . . . . 8  |-  ( -g `  S )  =  (
-g `  S )
20 eqid 2454 . . . . . . . 8  |-  ( -g `  T )  =  (
-g `  T )
2110, 19, 20ghmsub 15875 . . . . . . 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 1219 . . . . . 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 2454 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
2410, 16, 23ghmlin 15872 . . . . . . . 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 1219 . . . . . . 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 6216 . . . . . 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 2495 . . . . 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 15871 . . . . . . . . 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 5668 . . . . . . 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 15834 . . . . . . 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 1219 . . . . . 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 6065 . . . . . 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 1219 . . . . 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 2543 . . . 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 2914 . . 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 6208 . . . . . . . . 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 6219 . . . . . . . 8  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  T ) y ) ( -g `  T
) x )  =  ( ( ( F `
 z ) ( +g  `  T ) y ) ( -g `  T ) ( F `
 z ) ) )
4544eleq1d 2523 . . . . . . 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 2846 . . . . . 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 5956 . . . . 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 3029 . . . 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 6209 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  T
) y )  =  ( ( F `  z ) ( +g  `  T ) ( F `
 x ) ) )
5251oveq1d 6216 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  T ) y ) ( -g `  T
) ( F `  z ) )  =  ( ( ( F `
 z ) ( +g  `  T ) ( F `  x
) ) ( -g `  T ) ( F `
 z ) ) )
5352eleq1d 2523 . . . . . . 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 6067 . . . . . 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 2846 . . . 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 15835 . 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 1370    e. wcel 1758   A.wral 2799    C_ wss 3437   ran crn 4950   "cima 4952    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   Grpcgrp 15530   -gcsg 15533  SubGrpcsubg 15795  NrmSGrpcnsg 15796    GrpHom cghm 15864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-nsg 15799  df-ghm 15865
This theorem is referenced by: (None)
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