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Theorem cycsubgcyg 17030
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 2457 . 2  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2 eqid 2457 . 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 2457 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
74, 5, 6cycsubgcl 16354 . . . . 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 459 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  e.  (SubGrp `  G )
)
93, 8syl5eqel 2549 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  S  e.  (SubGrp `  G ) )
10 eqid 2457 . . . 4  |-  ( Gs  S )  =  ( Gs  S )
1110subggrp 16331 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( Gs  S
)  e.  Grp )
129, 11syl 16 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e.  Grp )
137simprd 463 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ran  (
x  e.  ZZ  |->  ( x  .x.  A ) ) )
1413, 3syl6eleqr 2556 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  S )
1510subgbas 16332 . . . 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 2547 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ( Base `  ( Gs  S ) ) )
1816eleq2d 2527 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( y  e.  S  <->  y  e.  ( Base `  ( Gs  S ) ) ) )
1918biimpar 485 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  (
Base `  ( Gs  S
) ) )  -> 
y  e.  S )
20 simpr 461 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  S )
2120, 3syl6eleq 2555 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
22 oveq1 6303 . . . . . . 7  |-  ( x  =  n  ->  (
x  .x.  A )  =  ( n  .x.  A ) )
2322cbvmptv 4548 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( n  e.  ZZ  |->  ( n  .x.  A ) )
24 ovex 6324 . . . . . 6  |-  ( n 
.x.  A )  e. 
_V
2523, 24elrnmpti 5263 . . . . 5  |-  ( y  e.  ran  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  <->  E. n  e.  ZZ  y  =  ( n  .x.  A ) )
2621, 25sylib 196 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  E. n  e.  ZZ  y  =  ( n  .x.  A ) )
279ad2antrr 725 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  S  e.  (SubGrp `  G )
)
28 simpr 461 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
2914ad2antrr 725 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  A  e.  S )
305, 10, 2subgmulg 16342 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  A  e.  S )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3127, 28, 29, 30syl3anc 1228 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3231eqeq2d 2471 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
y  =  ( n 
.x.  A )  <->  y  =  ( n (.g `  ( Gs  S ) ) A ) ) )
3332rexbidva 2965 . . . 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 210 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  E. n  e.  ZZ  y  =  ( n (.g `  ( Gs  S ) ) A ) )
3519, 34syldan 470 . 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 17017 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e. CycGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    |-> cmpt 4515   ran crn 5009   ` cfv 5594  (class class class)co 6296   ZZcz 10885   Basecbs 14644   ↾s cress 14645   Grpcgrp 16180  .gcmg 16183  SubGrpcsubg 16322  CycGrpccyg 17007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-mulg 16187  df-subg 16325  df-cyg 17008
This theorem is referenced by:  cycsubgcyg2  17031
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