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Theorem ghmrn 14947
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 2388 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2388 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 14938 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frn 5538 . . 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 5536 . . . . 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 14936 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
91grpbn0 14762 . . . . 5  |-  ( S  e.  Grp  ->  ( Base `  S )  =/=  (/) )
108, 9syl 16 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Base `  S )  =/=  (/) )
117, 10eqnetrd 2569 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  dom  F  =/=  (/) )
12 dm0rn0 5027 . . . 4  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2583 . . 3  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 189 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  =/=  (/) )
15 eqid 2388 . . . . . . . . . 10  |-  ( +g  `  S )  =  ( +g  `  S )
16 eqid 2388 . . . . . . . . . 10  |-  ( +g  `  T )  =  ( +g  `  T )
171, 15, 16ghmlin 14939 . . . . . . . . 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 5532 . . . . . . . . . . . 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 14746 . . . . . . . . . . 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 5807 . . . . . . . . . 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 2463 . . . . . . . 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 2732 . . . . . 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 6029 . . . . . . . . . 10  |-  ( b  =  ( F `  a )  ->  (
( F `  c
) ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) ( F `
 a ) ) )
2928eleq1d 2454 . . . . . . . . 9  |-  ( b  =  ( F `  a )  ->  (
( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F ) )
3029ralrn 5813 . . . . . . . 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 2388 . . . . . . 7  |-  ( inv g `  S )  =  ( inv g `  S )
35 eqid 2388 . . . . . . 7  |-  ( inv g `  T )  =  ( inv g `  T )
361, 34, 35ghminv 14941 . . . . . 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 14778 . . . . . . . 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 5807 . . . . . . 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 2463 . . . . 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 2733 . . 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 6028 . . . . . . . 8  |-  ( a  =  ( F `  c )  ->  (
a ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) b ) )
4645eleq1d 2454 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( a ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) b )  e.  ran  F ) )
4746ralbidv 2670 . . . . . 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 5669 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( inv g `  T ) `  a
)  =  ( ( inv g `  T
) `  ( F `  c ) ) )
4948eleq1d 2454 . . . . . 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 5813 . . . 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 14937 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
552, 16, 35issubg2 14887 . . 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 1649    e. wcel 1717    =/= wne 2551   A.wral 2650    C_ wss 3264   (/)c0 3572   dom cdm 4819   ran crn 4820    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   Grpcgrp 14613   inv gcminusg 14614  SubGrpcsubg 14866    GrpHom cghm 14931
This theorem is referenced by:  ghmima  14954  cayley  15040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-subg 14869  df-ghm 14932
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