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Theorem ghmcyg 15425
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 2380 . . . 4  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 15409 . . 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 2380 . . . 4  |-  (.g `  H
)  =  (.g `  H
)
7 ghmgrp2 14929 . . . . 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 5586 . . . . . 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 5803 . . . 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 5583 . . . . . . . . 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 5820 . . . . . . 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 6038 . . . . . . . 8  |-  ( m (.g `  G ) x )  e.  _V
2019rgenw 2709 . . . . . . 7  |-  A. m  e.  ZZ  ( m (.g `  G ) x )  e.  _V
21 oveq1 6020 . . . . . . . . 9  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
2221cbvmptv 4234 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  ( m  e.  ZZ  |->  ( m (.g `  G ) x ) )
23 fveq2 5661 . . . . . . . . 9  |-  ( z  =  ( m (.g `  G ) x )  ->  ( F `  z )  =  ( F `  ( m (.g `  G ) x ) ) )
2423eqeq2d 2391 . . . . . . . 8  |-  ( z  =  ( m (.g `  G ) x )  ->  ( y  =  ( F `  z
)  <->  y  =  ( F `  ( m (.g `  G ) x ) ) ) )
2522, 24rexrnmpt 5811 . . . . . . 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 14938 . . . . . . . 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 2391 . . . . . 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 2659 . . . . 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 15417 . . 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 2767 . 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 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   _Vcvv 2892    e. cmpt 4200   ran crn 4812   -->wf 5383   -onto->wfo 5385   ` cfv 5387  (class class class)co 6013   ZZcz 10207   Basecbs 13389   Grpcgrp 14605  .gcmg 14609    GrpHom cghm 14923  CycGrpccyg 15407
This theorem is referenced by:  giccyg  15429
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-seq 11244  df-0g 13647  df-mnd 14610  df-mhm 14658  df-grp 14732  df-minusg 14733  df-mulg 14735  df-ghm 14924  df-cyg 15408
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