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Theorem ghmrn 15011
Description: The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmrn  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  e.  (SubGrp `  T )
)

Proof of Theorem ghmrn
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2435 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 15002 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frn 5589 . . 3  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  ran  F  C_  ( Base `  T )
)
53, 4syl 16 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  C_  ( Base `  T )
)
6 fdm 5587 . . . . 5  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
73, 6syl 16 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  dom  F  =  ( Base `  S
) )
8 ghmgrp1 15000 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
91grpbn0 14826 . . . . 5  |-  ( S  e.  Grp  ->  ( Base `  S )  =/=  (/) )
108, 9syl 16 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Base `  S )  =/=  (/) )
117, 10eqnetrd 2616 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  dom  F  =/=  (/) )
12 dm0rn0 5078 . . . 4  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2630 . . 3  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 189 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  =/=  (/) )
15 eqid 2435 . . . . . . . . . 10  |-  ( +g  `  S )  =  ( +g  `  S )
16 eqid 2435 . . . . . . . . . 10  |-  ( +g  `  T )  =  ( +g  `  T )
171, 15, 16ghmlin 15003 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( F `  ( c ( +g  `  S ) a ) )  =  ( ( F `  c ) ( +g  `  T
) ( F `  a ) ) )
18 ffn 5583 . . . . . . . . . . . 12  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
193, 18syl 16 . . . . . . . . . . 11  |-  ( F  e.  ( S  GrpHom  T )  ->  F  Fn  ( Base `  S )
)
20193ad2ant1 978 . . . . . . . . . 10  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  F  Fn  ( Base `  S )
)
211, 15grpcl 14810 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  c  e.  ( Base `  S )  /\  a  e.  ( Base `  S
) )  ->  (
c ( +g  `  S
) a )  e.  ( Base `  S
) )
228, 21syl3an1 1217 . . . . . . . . . 10  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( c
( +g  `  S ) a )  e.  (
Base `  S )
)
23 fnfvelrn 5859 . . . . . . . . . 10  |-  ( ( F  Fn  ( Base `  S )  /\  (
c ( +g  `  S
) a )  e.  ( Base `  S
) )  ->  ( F `  ( c
( +g  `  S ) a ) )  e. 
ran  F )
2420, 22, 23syl2anc 643 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( F `  ( c ( +g  `  S ) a ) )  e.  ran  F
)
2517, 24eqeltrrd 2510 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F )
26253expia 1155 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
a  e.  ( Base `  S )  ->  (
( F `  c
) ( +g  `  T
) ( F `  a ) )  e. 
ran  F ) )
2726ralrimiv 2780 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  A. a  e.  ( Base `  S
) ( ( F `
 c ) ( +g  `  T ) ( F `  a
) )  e.  ran  F )
28 oveq2 6081 . . . . . . . . . 10  |-  ( b  =  ( F `  a )  ->  (
( F `  c
) ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) ( F `
 a ) ) )
2928eleq1d 2501 . . . . . . . . 9  |-  ( b  =  ( F `  a )  ->  (
( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F ) )
3029ralrn 5865 . . . . . . . 8  |-  ( F  Fn  ( Base `  S
)  ->  ( A. b  e.  ran  F ( ( F `  c
) ( +g  `  T
) b )  e. 
ran  F  <->  A. a  e.  (
Base `  S )
( ( F `  c ) ( +g  `  T ) ( F `
 a ) )  e.  ran  F ) )
3119, 30syl 16 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  ( A. b  e.  ran  F ( ( F `  c
) ( +g  `  T
) b )  e. 
ran  F  <->  A. a  e.  (
Base `  S )
( ( F `  c ) ( +g  `  T ) ( F `
 a ) )  e.  ran  F ) )
3231adantr 452 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  <->  A. a  e.  ( Base `  S
) ( ( F `
 c ) ( +g  `  T ) ( F `  a
) )  e.  ran  F ) )
3327, 32mpbird 224 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F )
34 eqid 2435 . . . . . . 7  |-  ( inv g `  S )  =  ( inv g `  S )
35 eqid 2435 . . . . . . 7  |-  ( inv g `  T )  =  ( inv g `  T )
361, 34, 35ghminv 15005 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 c ) )  =  ( ( inv g `  T ) `
 ( F `  c ) ) )
3719adantr 452 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  F  Fn  ( Base `  S
) )
381, 34grpinvcl 14842 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  c  e.  ( Base `  S ) )  -> 
( ( inv g `  S ) `  c
)  e.  ( Base `  S ) )
398, 38sylan 458 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
( inv g `  S ) `  c
)  e.  ( Base `  S ) )
40 fnfvelrn 5859 . . . . . . 7  |-  ( ( F  Fn  ( Base `  S )  /\  (
( inv g `  S ) `  c
)  e.  ( Base `  S ) )  -> 
( F `  (
( inv g `  S ) `  c
) )  e.  ran  F )
4137, 39, 40syl2anc 643 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 c ) )  e.  ran  F )
4236, 41eqeltrrd 2510 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
)
4333, 42jca 519 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  /\  ( ( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) )
4443ralrimiva 2781 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  A. c  e.  ( Base `  S
) ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) )
45 oveq1 6080 . . . . . . . 8  |-  ( a  =  ( F `  c )  ->  (
a ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) b ) )
4645eleq1d 2501 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( a ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) b )  e.  ran  F ) )
4746ralbidv 2717 . . . . . 6  |-  ( a  =  ( F `  c )  ->  ( A. b  e.  ran  F ( a ( +g  `  T ) b )  e.  ran  F  <->  A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F ) )
48 fveq2 5720 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( inv g `  T ) `  a
)  =  ( ( inv g `  T
) `  ( F `  c ) ) )
4948eleq1d 2501 . . . . . 6  |-  ( a  =  ( F `  c )  ->  (
( ( inv g `  T ) `  a
)  e.  ran  F  <->  ( ( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) )
5047, 49anbi12d 692 . . . . 5  |-  ( a  =  ( F `  c )  ->  (
( A. b  e. 
ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
)  <->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5150ralrn 5865 . . . 4  |-  ( F  Fn  ( Base `  S
)  ->  ( A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T ) b )  e.  ran  F  /\  ( ( inv g `  T ) `  a
)  e.  ran  F
)  <->  A. c  e.  (
Base `  S )
( A. b  e. 
ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5219, 51syl 16 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T ) b )  e.  ran  F  /\  ( ( inv g `  T ) `  a
)  e.  ran  F
)  <->  A. c  e.  (
Base `  S )
( A. b  e. 
ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5344, 52mpbird 224 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
) )
54 ghmgrp2 15001 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
552, 16, 35issubg2 14951 . . 3  |-  ( T  e.  Grp  ->  ( ran  F  e.  (SubGrp `  T )  <->  ( ran  F 
C_  ( Base `  T
)  /\  ran  F  =/=  (/)  /\  A. a  e. 
ran  F ( A. b  e.  ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
) ) ) )
5654, 55syl 16 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( ran  F  e.  (SubGrp `  T
)  <->  ( ran  F  C_  ( Base `  T
)  /\  ran  F  =/=  (/)  /\  A. a  e. 
ran  F ( A. b  e.  ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
) ) ) )
575, 14, 53, 56mpbir3and 1137 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  e.  (SubGrp `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312   (/)c0 3620   dom cdm 4870   ran crn 4871    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   inv gcminusg 14678  SubGrpcsubg 14930    GrpHom cghm 14995
This theorem is referenced by:  ghmima  15018  cayley  15104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-subg 14933  df-ghm 14996
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