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Theorem ghmcyg 16365
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 2441 . . . 4  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 16349 . . 3  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )
43simprbi 461 . 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 2441 . . . 4  |-  (.g `  H
)  =  (.g `  H
)
7 ghmgrp2 15743 . . . . 5  |-  ( F  e.  ( G  GrpHom  H )  ->  H  e.  Grp )
87ad2antrr 720 . . . 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 5617 . . . . . 6  |-  ( F : B -onto-> C  ->  F : B --> C )
109ad2antlr 721 . . . . 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 750 . . . . 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 5841 . . . 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 749 . . . . . . . 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 5614 . . . . . . . . 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 723 . . . . . . . 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 232 . . . . . . 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 5859 . . . . . . 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 468 . . . . . 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 6115 . . . . . . . 8  |-  ( m (.g `  G ) x )  e.  _V
2019rgenw 2781 . . . . . . 7  |-  A. m  e.  ZZ  ( m (.g `  G ) x )  e.  _V
21 oveq1 6097 . . . . . . . . 9  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
2221cbvmptv 4380 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  ( m  e.  ZZ  |->  ( m (.g `  G ) x ) )
23 fveq2 5688 . . . . . . . . 9  |-  ( z  =  ( m (.g `  G ) x )  ->  ( F `  z )  =  ( F `  ( m (.g `  G ) x ) ) )
2423eqeq2d 2452 . . . . . . . 8  |-  ( z  =  ( m (.g `  G ) x )  ->  ( y  =  ( F `  z
)  <->  y  =  ( F `  ( m (.g `  G ) x ) ) ) )
2522, 24rexrnmpt 5850 . . . . . . 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 5 . . . . . 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 196 . . . . 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 760 . . . . . . . 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 458 . . . . . . . 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 720 . . . . . . . 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 15752 . . . . . . . 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 1213 . . . . . . 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 2452 . . . . . 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 2730 . . . . 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 210 . . . 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 16357 . . 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 2840 . 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 32 1  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  F : B -onto-> C )  ->  ( G  e. CycGrp  ->  H  e. CycGrp
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    e. cmpt 4347   ran crn 4837   -->wf 5411   -onto->wfo 5413   ` cfv 5415  (class class class)co 6090   ZZcz 10642   Basecbs 14170   Grpcgrp 15406  .gcmg 15410    GrpHom cghm 15737  CycGrpccyg 16347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-0g 14376  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-mulg 15541  df-ghm 15738  df-cyg 16348
This theorem is referenced by:  giccyg  16369
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