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Theorem ghmrn 15781
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 2443 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2443 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 15772 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frn 5586 . . 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 5584 . . . . 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 15770 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
91grpbn0 15588 . . . . 5  |-  ( S  e.  Grp  ->  ( Base `  S )  =/=  (/) )
108, 9syl 16 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Base `  S )  =/=  (/) )
117, 10eqnetrd 2654 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  dom  F  =/=  (/) )
12 dm0rn0 5077 . . . 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 2443 . . . . . . . . . 10  |-  ( +g  `  S )  =  ( +g  `  S )
16 eqid 2443 . . . . . . . . . 10  |-  ( +g  `  T )  =  ( +g  `  T )
171, 15, 16ghmlin 15773 . . . . . . . . 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 5580 . . . . . . . . . . . 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 15572 . . . . . . . . . . 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 5861 . . . . . . . . . 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 2518 . . . . . . . 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 2819 . . . . . 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 6120 . . . . . . . . . 10  |-  ( b  =  ( F `  a )  ->  (
( F `  c
) ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) ( F `
 a ) ) )
2928eleq1d 2509 . . . . . . . . 9  |-  ( b  =  ( F `  a )  ->  (
( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F ) )
3029ralrn 5867 . . . . . . . 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 2443 . . . . . . 7  |-  ( invg `  S )  =  ( invg `  S )
35 eqid 2443 . . . . . . 7  |-  ( invg `  T )  =  ( invg `  T )
361, 34, 35ghminv 15775 . . . . . 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 15604 . . . . . . . 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 5861 . . . . . . 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 2518 . . . . 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 2820 . . 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 6119 . . . . . . . 8  |-  ( a  =  ( F `  c )  ->  (
a ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) b ) )
4645eleq1d 2509 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( a ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) b )  e.  ran  F ) )
4746ralbidv 2756 . . . . . 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 5712 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( invg `  T ) `  a
)  =  ( ( invg `  T
) `  ( F `  c ) ) )
4948eleq1d 2509 . . . . . 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 5867 . . . 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 15771 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
552, 16, 35issubg2 15717 . . 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 2620   A.wral 2736    C_ wss 3349   (/)c0 3658   dom cdm 4861   ran crn 4862    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   Grpcgrp 15431   invgcminusg 15432  SubGrpcsubg 15696    GrpHom cghm 15765
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-subg 15699  df-ghm 15766
This theorem is referenced by:  ghmima  15788  cayley  15940
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