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Theorem ghmcyg 17465
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 2429 . . . 4  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 17449 . . 3  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )
43simprbi 465 . 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 2429 . . . 4  |-  (.g `  H
)  =  (.g `  H
)
7 ghmgrp2 16837 . . . . 5  |-  ( F  e.  ( G  GrpHom  H )  ->  H  e.  Grp )
87ad2antrr 730 . . . 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 5810 . . . . . 6  |-  ( F : B -onto-> C  ->  F : B --> C )
109ad2antlr 731 . . . . 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 762 . . . . 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 6038 . . . 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 760 . . . . . . . 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 5807 . . . . . . . . 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 733 . . . . . . . 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 235 . . . . . . 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 6056 . . . . . . 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 473 . . . . . 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 6333 . . . . . . . 8  |-  ( m (.g `  G ) x )  e.  _V
2019rgenw 2793 . . . . . . 7  |-  A. m  e.  ZZ  ( m (.g `  G ) x )  e.  _V
21 oveq1 6312 . . . . . . . . 9  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
2221cbvmptv 4518 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  ( m  e.  ZZ  |->  ( m (.g `  G ) x ) )
23 fveq2 5881 . . . . . . . . 9  |-  ( z  =  ( m (.g `  G ) x )  ->  ( F `  z )  =  ( F `  ( m (.g `  G ) x ) ) )
2423eqeq2d 2443 . . . . . . . 8  |-  ( z  =  ( m (.g `  G ) x )  ->  ( y  =  ( F `  z
)  <->  y  =  ( F `  ( m (.g `  G ) x ) ) ) )
2522, 24rexrnmpt 6047 . . . . . . 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 199 . . . . 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 774 . . . . . . . 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 462 . . . . . . . 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 730 . . . . . . . 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 16846 . . . . . . . 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 1264 . . . . . . 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 2443 . . . . . 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 2943 . . . . 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 213 . . . 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 17457 . . 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 2925 . 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 33 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087    |-> cmpt 4484   ran crn 4855   -->wf 5597   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   ZZcz 10937   Basecbs 15084   Grpcgrp 16620  .gcmg 16623    GrpHom cghm 16831  CycGrpccyg 17447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-seq 12211  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-grp 16624  df-minusg 16625  df-mulg 16627  df-ghm 16832  df-cyg 17448
This theorem is referenced by:  giccyg  17469
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