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Theorem ghmcyg 16698
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 2467 . . . 4  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 16682 . . 3  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  B ) )
43simprbi 464 . 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 2467 . . . 4  |-  (.g `  H
)  =  (.g `  H
)
7 ghmgrp2 16072 . . . . 5  |-  ( F  e.  ( G  GrpHom  H )  ->  H  e.  Grp )
87ad2antrr 725 . . . 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 5794 . . . . . 6  |-  ( F : B -onto-> C  ->  F : B --> C )
109ad2antlr 726 . . . . 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 755 . . . . 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 6021 . . . 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 754 . . . . . . . 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 5791 . . . . . . . . 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 728 . . . . . . . 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 6039 . . . . . . 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 471 . . . . . 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 6308 . . . . . . . 8  |-  ( m (.g `  G ) x )  e.  _V
2019rgenw 2825 . . . . . . 7  |-  A. m  e.  ZZ  ( m (.g `  G ) x )  e.  _V
21 oveq1 6290 . . . . . . . . 9  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
2221cbvmptv 4538 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n (.g `  G ) x ) )  =  ( m  e.  ZZ  |->  ( m (.g `  G ) x ) )
23 fveq2 5865 . . . . . . . . 9  |-  ( z  =  ( m (.g `  G ) x )  ->  ( F `  z )  =  ( F `  ( m (.g `  G ) x ) ) )
2423eqeq2d 2481 . . . . . . . 8  |-  ( z  =  ( m (.g `  G ) x )  ->  ( y  =  ( F `  z
)  <->  y  =  ( F `  ( m (.g `  G ) x ) ) ) )
2522, 24rexrnmpt 6030 . . . . . . 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 765 . . . . . . . 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 461 . . . . . . . 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 725 . . . . . . . 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 16081 . . . . . . . 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 1228 . . . . . . 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 2481 . . . . . 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 2970 . . . . 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 16690 . . 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 2956 . 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    |-> cmpt 4505   ran crn 5000   -->wf 5583   -onto->wfo 5585   ` cfv 5587  (class class class)co 6283   ZZcz 10863   Basecbs 14489   Grpcgrp 15726  .gcmg 15730    GrpHom cghm 16066  CycGrpccyg 16680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-seq 12075  df-0g 14696  df-mnd 15731  df-mhm 15783  df-grp 15864  df-minusg 15865  df-mulg 15867  df-ghm 16067  df-cyg 16681
This theorem is referenced by:  giccyg  16702
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