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Theorem ghmcyg 15497
Description: The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
cygctb.1  |-  B  =  ( Base `  G
)
ghmcyg.1  |-  C  =  ( Base `  H
)
Assertion
Ref Expression
ghmcyg  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  F : B -onto-> C )  ->  ( G  e. CycGrp  ->  H  e. CycGrp
) )

Proof of Theorem ghmcyg
Dummy variables  m  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cygctb.1 . . . 4  |-  B  =  ( Base `  G
)
2 eqid 2435 . . . 4  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 15481 . . 3  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )
43simprbi 451 . 2  |-  ( G  e. CycGrp  ->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B )
5 ghmcyg.1 . . . 4  |-  C  =  ( Base `  H
)
6 eqid 2435 . . . 4  |-  (.g `  H
)  =  (.g `  H
)
7 ghmgrp2 15001 . . . . 5  |-  ( F  e.  ( G  GrpHom  H )  ->  H  e.  Grp )
87ad2antrr 707 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  H  e.  Grp )
9 fof 5645 . . . . . 6  |-  ( F : B -onto-> C  ->  F : B --> C )
109ad2antlr 708 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  F : B --> C )
11 simprl 733 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  x  e.  B )
1210, 11ffvelrnd 5863 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  ( F `  x )  e.  C )
13 simplr 732 . . . . . . . 8  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  F : B -onto-> C )
14 foeq2 5642 . . . . . . . . 9  |-  ( ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B  ->  ( F : ran  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) ) -onto-> C  <->  F : B -onto-> C ) )
1514ad2antll 710 . . . . . . . 8  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  ( F : ran  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) ) -onto-> C  <->  F : B -onto-> C ) )
1613, 15mpbird 224 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  F : ran  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) ) -onto-> C )
17 foelrn 5880 . . . . . . 7  |-  ( ( F : ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) ) -onto-> C  /\  y  e.  C )  ->  E. z  e.  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) ) y  =  ( F `
 z ) )
1816, 17sylan 458 . . . . . 6  |-  ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  ->  E. z  e.  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) ) y  =  ( F `  z ) )
19 ovex 6098 . . . . . . . 8  |-  ( m (.g `  G ) x )  e.  _V
2019rgenw 2765 . . . . . . 7  |-  A. m  e.  ZZ  ( m (.g `  G ) x )  e.  _V
21 oveq1 6080 . . . . . . . . 9  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
2221cbvmptv 4292 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  ( m  e.  ZZ  |->  ( m (.g `  G ) x ) )
23 fveq2 5720 . . . . . . . . 9  |-  ( z  =  ( m (.g `  G ) x )  ->  ( F `  z )  =  ( F `  ( m (.g `  G ) x ) ) )
2423eqeq2d 2446 . . . . . . . 8  |-  ( z  =  ( m (.g `  G ) x )  ->  ( y  =  ( F `  z
)  <->  y  =  ( F `  ( m (.g `  G ) x ) ) ) )
2522, 24rexrnmpt 5871 . . . . . . 7  |-  ( A. m  e.  ZZ  (
m (.g `  G ) x )  e.  _V  ->  ( E. z  e.  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) ) y  =  ( F `
 z )  <->  E. m  e.  ZZ  y  =  ( F `  ( m (.g `  G ) x ) ) ) )
2620, 25ax-mp 8 . . . . . 6  |-  ( E. z  e.  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) ) y  =  ( F `  z
)  <->  E. m  e.  ZZ  y  =  ( F `  ( m (.g `  G
) x ) ) )
2718, 26sylib 189 . . . . 5  |-  ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  ->  E. m  e.  ZZ  y  =  ( F `  ( m (.g `  G ) x ) ) )
28 simp-4l 743 . . . . . . . 8  |-  ( ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  /\  m  e.  ZZ )  ->  F  e.  ( G  GrpHom  H ) )
29 simpr 448 . . . . . . . 8  |-  ( ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  /\  m  e.  ZZ )  ->  m  e.  ZZ )
3011ad2antrr 707 . . . . . . . 8  |-  ( ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  /\  m  e.  ZZ )  ->  x  e.  B )
311, 2, 6ghmmulg 15010 . . . . . . . 8  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  m  e.  ZZ  /\  x  e.  B )  ->  ( F `  ( m
(.g `  G ) x ) )  =  ( m (.g `  H ) ( F `  x ) ) )
3228, 29, 30, 31syl3anc 1184 . . . . . . 7  |-  ( ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  /\  m  e.  ZZ )  ->  ( F `
 ( m (.g `  G ) x ) )  =  ( m (.g `  H ) ( F `  x ) ) )
3332eqeq2d 2446 . . . . . 6  |-  ( ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  /\  m  e.  ZZ )  ->  ( y  =  ( F `  ( m (.g `  G
) x ) )  <-> 
y  =  ( m (.g `  H ) ( F `  x ) ) ) )
3433rexbidva 2714 . . . . 5  |-  ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  ->  ( E. m  e.  ZZ  y  =  ( F `  ( m (.g `  G
) x ) )  <->  E. m  e.  ZZ  y  =  ( m
(.g `  H ) ( F `  x ) ) ) )
3527, 34mpbid 202 . . . 4  |-  ( ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  (
x  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )  /\  y  e.  C
)  ->  E. m  e.  ZZ  y  =  ( m (.g `  H ) ( F `  x ) ) )
365, 6, 8, 12, 35iscygd 15489 . . 3  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  /\  ( x  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  B ) )  ->  H  e. CycGrp )
3736rexlimdvaa 2823 . 2  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  F : B -onto-> C )  ->  ( E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B  ->  H  e. CycGrp ) )
384, 37syl5 30 1  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  F : B -onto-> C )  ->  ( G  e. CycGrp  ->  H  e. CycGrp
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    e. cmpt 4258   ran crn 4871   -->wf 5442   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073   ZZcz 10274   Basecbs 13461   Grpcgrp 14677  .gcmg 14681    GrpHom cghm 14995  CycGrpccyg 15479
This theorem is referenced by:  giccyg  15501
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-inf2 7588  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-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  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-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-mulg 14807  df-ghm 14996  df-cyg 15480
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