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Theorem cycsubgcyg 16370
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 2441 . 2  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2 eqid 2441 . 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 2441 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
74, 5, 6cycsubgcl 15700 . . . . 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 456 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  e.  (SubGrp `  G )
)
93, 8syl5eqel 2525 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  S  e.  (SubGrp `  G ) )
10 eqid 2441 . . . 4  |-  ( Gs  S )  =  ( Gs  S )
1110subggrp 15677 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( Gs  S
)  e.  Grp )
129, 11syl 16 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e.  Grp )
137simprd 460 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ran  (
x  e.  ZZ  |->  ( x  .x.  A ) ) )
1413, 3syl6eleqr 2532 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  S )
1510subgbas 15678 . . . 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 2517 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ( Base `  ( Gs  S ) ) )
1816eleq2d 2508 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( y  e.  S  <->  y  e.  ( Base `  ( Gs  S ) ) ) )
1918biimpar 482 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  (
Base `  ( Gs  S
) ) )  -> 
y  e.  S )
20 simpr 458 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  S )
2120, 3syl6eleq 2531 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
22 oveq1 6097 . . . . . . 7  |-  ( x  =  n  ->  (
x  .x.  A )  =  ( n  .x.  A ) )
2322cbvmptv 4380 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( n  e.  ZZ  |->  ( n  .x.  A ) )
24 ovex 6115 . . . . . 6  |-  ( n 
.x.  A )  e. 
_V
2523, 24elrnmpti 5086 . . . . 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 720 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  S  e.  (SubGrp `  G )
)
28 simpr 458 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
2914ad2antrr 720 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  A  e.  S )
305, 10, 2subgmulg 15688 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  A  e.  S )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3127, 28, 29, 30syl3anc 1213 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
n  .x.  A )  =  ( n (.g `  ( Gs  S ) ) A ) )
3231eqeq2d 2452 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
y  =  ( n 
.x.  A )  <->  y  =  ( n (.g `  ( Gs  S ) ) A ) ) )
3332rexbidva 2730 . . . 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 467 . 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 16357 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e. CycGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714    e. cmpt 4347   ran crn 4837   ` cfv 5415  (class class class)co 6090   ZZcz 10642   Basecbs 14170   ↾s cress 14171   Grpcgrp 15406  .gcmg 15410  SubGrpcsubg 15668  CycGrpccyg 16347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-mulg 15541  df-subg 15671  df-cyg 16348
This theorem is referenced by:  cycsubgcyg2  16371
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