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Theorem cycsubgcyg 15465
Description: The cyclic subgroup generated by  A is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
cycsubgcyg.x  |-  X  =  ( Base `  G
)
cycsubgcyg.t  |-  .x.  =  (.g
`  G )
cycsubgcyg.s  |-  S  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )
Assertion
Ref Expression
cycsubgcyg  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e. CycGrp )
Distinct variable groups:    x, A    x, G    x,  .x.    x, X
Allowed substitution hint:    S( x)

Proof of Theorem cycsubgcyg
Dummy variables  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2 eqid 2404 . 2  |-  (.g `  ( Gs  S ) )  =  (.g `  ( Gs  S ) )
3 cycsubgcyg.s . . . 4  |-  S  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )
4 cycsubgcyg.x . . . . . 6  |-  X  =  ( Base `  G
)
5 cycsubgcyg.t . . . . . 6  |-  .x.  =  (.g
`  G )
6 eqid 2404 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
74, 5, 6cycsubgcl 14921 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ran  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  e.  (SubGrp `  G
)  /\  A  e.  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) ) )
87simpld 446 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  e.  (SubGrp `  G )
)
93, 8syl5eqel 2488 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  S  e.  (SubGrp `  G ) )
10 eqid 2404 . . . 4  |-  ( Gs  S )  =  ( Gs  S )
1110subggrp 14902 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( Gs  S
)  e.  Grp )
129, 11syl 16 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e.  Grp )
137simprd 450 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ran  (
x  e.  ZZ  |->  ( x  .x.  A ) ) )
1413, 3syl6eleqr 2495 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  S )
1510subgbas 14903 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  ( Gs  S
) ) )
169, 15syl 16 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  S  =  ( Base `  ( Gs  S ) ) )
1714, 16eleqtrd 2480 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ( Base `  ( Gs  S ) ) )
1816eleq2d 2471 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( y  e.  S  <->  y  e.  ( Base `  ( Gs  S ) ) ) )
1918biimpar 472 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  (
Base `  ( Gs  S
) ) )  -> 
y  e.  S )
20 simpr 448 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  S )
2120, 3syl6eleq 2494 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
22 oveq1 6047 . . . . . . 7  |-  ( x  =  n  ->  (
x  .x.  A )  =  ( n  .x.  A ) )
2322cbvmptv 4260 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( n  e.  ZZ  |->  ( n  .x.  A ) )
24 ovex 6065 . . . . . 6  |-  ( n 
.x.  A )  e. 
_V
2523, 24elrnmpti 5080 . . . . 5  |-  ( y  e.  ran  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  <->  E. n  e.  ZZ  y  =  ( n  .x.  A ) )
2621, 25sylib 189 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  E. n  e.  ZZ  y  =  ( n  .x.  A ) )
279ad2antrr 707 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  S  e.  (SubGrp `  G )
)
28 simpr 448 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
2914ad2antrr 707 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  A  e.  S )
305, 10, 2subgmulg 14913 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  A  e.  S )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3127, 28, 29, 30syl3anc 1184 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3231eqeq2d 2415 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
y  =  ( n 
.x.  A )  <->  y  =  ( n (.g `  ( Gs  S ) ) A ) ) )
3332rexbidva 2683 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  ( E. n  e.  ZZ  y  =  ( n  .x.  A )  <->  E. n  e.  ZZ  y  =  ( n (.g `  ( Gs  S ) ) A ) ) )
3426, 33mpbid 202 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  E. n  e.  ZZ  y  =  ( n (.g `  ( Gs  S ) ) A ) )
3519, 34syldan 457 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  (
Base `  ( Gs  S
) ) )  ->  E. n  e.  ZZ  y  =  ( n
(.g `  ( Gs  S ) ) A ) )
361, 2, 12, 17, 35iscygd 15452 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e. CycGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    e. cmpt 4226   ran crn 4838   ` cfv 5413  (class class class)co 6040   ZZcz 10238   Basecbs 13424   ↾s cress 13425   Grpcgrp 14640  .gcmg 14644  SubGrpcsubg 14893  CycGrpccyg 15442
This theorem is referenced by:  cycsubgcyg2  15466
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-mulg 14770  df-subg 14896  df-cyg 15443
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