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Theorem cycsubgcyg 15187
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 2283 . 2  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2 eqid 2283 . 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 2283 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
74, 5, 6cycsubgcl 14643 . . . . 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 445 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  e.  (SubGrp `  G )
)
93, 8syl5eqel 2367 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  S  e.  (SubGrp `  G ) )
10 eqid 2283 . . . 4  |-  ( Gs  S )  =  ( Gs  S )
1110subggrp 14624 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( Gs  S
)  e.  Grp )
129, 11syl 15 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e.  Grp )
137simprd 449 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ran  (
x  e.  ZZ  |->  ( x  .x.  A ) ) )
1413, 3syl6eleqr 2374 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  S )
1510subgbas 14625 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  ( Gs  S
) ) )
169, 15syl 15 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  S  =  ( Base `  ( Gs  S ) ) )
1714, 16eleqtrd 2359 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ( Base `  ( Gs  S ) ) )
1816eleq2d 2350 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( y  e.  S  <->  y  e.  ( Base `  ( Gs  S ) ) ) )
1918biimpar 471 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  (
Base `  ( Gs  S
) ) )  -> 
y  e.  S )
20 simpr 447 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  S )
2120, 3syl6eleq 2373 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
22 oveq1 5865 . . . . . . 7  |-  ( x  =  n  ->  (
x  .x.  A )  =  ( n  .x.  A ) )
2322cbvmptv 4111 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( n  e.  ZZ  |->  ( n  .x.  A ) )
24 ovex 5883 . . . . . 6  |-  ( n 
.x.  A )  e. 
_V
2523, 24elrnmpti 4930 . . . . 5  |-  ( y  e.  ran  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  <->  E. n  e.  ZZ  y  =  ( n  .x.  A ) )
2621, 25sylib 188 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  E. n  e.  ZZ  y  =  ( n  .x.  A ) )
279ad2antrr 706 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  S  e.  (SubGrp `  G )
)
28 simpr 447 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
2914ad2antrr 706 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  A  e.  S )
305, 10, 2subgmulg 14635 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  A  e.  S )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3127, 28, 29, 30syl3anc 1182 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3231eqeq2d 2294 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
y  =  ( n 
.x.  A )  <->  y  =  ( n (.g `  ( Gs  S ) ) A ) ) )
3332rexbidva 2560 . . . 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 201 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  E. n  e.  ZZ  y  =  ( n (.g `  ( Gs  S ) ) A ) )
3519, 34syldan 456 . 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 15174 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e. CycGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   ZZcz 10024   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362  .gcmg 14366  SubGrpcsubg 14615  CycGrpccyg 15164
This theorem is referenced by:  cycsubgcyg2  15188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-cyg 15165
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