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Theorem ghmrn 15749
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 2437 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2437 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 15740 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frn 5558 . . 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 5556 . . . . 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 15738 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
91grpbn0 15556 . . . . 5  |-  ( S  e.  Grp  ->  ( Base `  S )  =/=  (/) )
108, 9syl 16 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Base `  S )  =/=  (/) )
117, 10eqnetrd 2620 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  dom  F  =/=  (/) )
12 dm0rn0 5048 . . . 4  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2634 . . 3  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 196 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  =/=  (/) )
15 eqid 2437 . . . . . . . . . 10  |-  ( +g  `  S )  =  ( +g  `  S )
16 eqid 2437 . . . . . . . . . 10  |-  ( +g  `  T )  =  ( +g  `  T )
171, 15, 16ghmlin 15741 . . . . . . . . 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 5552 . . . . . . . . . . . 12  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
193, 18syl 16 . . . . . . . . . . 11  |-  ( F  e.  ( S  GrpHom  T )  ->  F  Fn  ( Base `  S )
)
20193ad2ant1 1009 . . . . . . . . . 10  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  F  Fn  ( Base `  S )
)
211, 15grpcl 15540 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  c  e.  ( Base `  S )  /\  a  e.  ( Base `  S
) )  ->  (
c ( +g  `  S
) a )  e.  ( Base `  S
) )
228, 21syl3an1 1251 . . . . . . . . . 10  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( c
( +g  `  S ) a )  e.  (
Base `  S )
)
23 fnfvelrn 5833 . . . . . . . . . 10  |-  ( ( F  Fn  ( Base `  S )  /\  (
c ( +g  `  S
) a )  e.  ( Base `  S
) )  ->  ( F `  ( c
( +g  `  S ) a ) )  e. 
ran  F )
2420, 22, 23syl2anc 661 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( F `  ( c ( +g  `  S ) a ) )  e.  ran  F
)
2517, 24eqeltrrd 2512 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F )
26253expia 1189 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
a  e.  ( Base `  S )  ->  (
( F `  c
) ( +g  `  T
) ( F `  a ) )  e. 
ran  F ) )
2726ralrimiv 2792 . . . . . 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 6094 . . . . . . . . . 10  |-  ( b  =  ( F `  a )  ->  (
( F `  c
) ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) ( F `
 a ) ) )
2928eleq1d 2503 . . . . . . . . 9  |-  ( b  =  ( F `  a )  ->  (
( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F ) )
3029ralrn 5839 . . . . . . . 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 465 . . . . . 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 232 . . . . 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 2437 . . . . . . 7  |-  ( invg `  S )  =  ( invg `  S )
35 eqid 2437 . . . . . . 7  |-  ( invg `  T )  =  ( invg `  T )
361, 34, 35ghminv 15743 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( F `  ( ( invg `  S ) `
 c ) )  =  ( ( invg `  T ) `
 ( F `  c ) ) )
3719adantr 465 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  F  Fn  ( Base `  S
) )
381, 34grpinvcl 15572 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  c  e.  ( Base `  S ) )  -> 
( ( invg `  S ) `  c
)  e.  ( Base `  S ) )
398, 38sylan 471 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
( invg `  S ) `  c
)  e.  ( Base `  S ) )
40 fnfvelrn 5833 . . . . . . 7  |-  ( ( F  Fn  ( Base `  S )  /\  (
( invg `  S ) `  c
)  e.  ( Base `  S ) )  -> 
( F `  (
( invg `  S ) `  c
) )  e.  ran  F )
4137, 39, 40syl2anc 661 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( F `  ( ( invg `  S ) `
 c ) )  e.  ran  F )
4236, 41eqeltrrd 2512 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
( invg `  T ) `  ( F `  c )
)  e.  ran  F
)
4333, 42jca 532 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  /\  ( ( invg `  T ) `  ( F `  c )
)  e.  ran  F
) )
4443ralrimiva 2793 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  A. c  e.  ( Base `  S
) ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( invg `  T ) `  ( F `  c )
)  e.  ran  F
) )
45 oveq1 6093 . . . . . . . 8  |-  ( a  =  ( F `  c )  ->  (
a ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) b ) )
4645eleq1d 2503 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( a ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) b )  e.  ran  F ) )
4746ralbidv 2729 . . . . . 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 5684 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( invg `  T ) `  a
)  =  ( ( invg `  T
) `  ( F `  c ) ) )
4948eleq1d 2503 . . . . . 6  |-  ( a  =  ( F `  c )  ->  (
( ( invg `  T ) `  a
)  e.  ran  F  <->  ( ( invg `  T ) `  ( F `  c )
)  e.  ran  F
) )
5047, 49anbi12d 710 . . . . 5  |-  ( a  =  ( F `  c )  ->  (
( A. b  e. 
ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( invg `  T ) `  a
)  e.  ran  F
)  <->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( invg `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5150ralrn 5839 . . . 4  |-  ( F  Fn  ( Base `  S
)  ->  ( A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T ) b )  e.  ran  F  /\  ( ( invg `  T ) `  a
)  e.  ran  F
)  <->  A. c  e.  (
Base `  S )
( A. b  e. 
ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( invg `  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  /\  ( ( invg `  T ) `  a
)  e.  ran  F
)  <->  A. c  e.  (
Base `  S )
( A. b  e. 
ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( invg `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5344, 52mpbird 232 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( invg `  T ) `  a
)  e.  ran  F
) )
54 ghmgrp2 15739 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
552, 16, 35issubg2 15685 . . 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  /\  (
( invg `  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  /\  (
( invg `  T ) `  a
)  e.  ran  F
) ) ) )
575, 14, 53, 56mpbir3and 1171 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  e.  (SubGrp `  T )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2600   A.wral 2709    C_ wss 3321   (/)c0 3630   dom cdm 4832   ran crn 4833    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   Grpcgrp 15402   invgcminusg 15403  SubGrpcsubg 15664    GrpHom cghm 15733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15534  df-minusg 15535  df-subg 15667  df-ghm 15734
This theorem is referenced by:  ghmima  15756  cayley  15908
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