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Theorem cycsubgcyg 15203
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 2296 . 2  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2 eqid 2296 . 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 2296 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
74, 5, 6cycsubgcl 14659 . . . . 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 2380 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  S  e.  (SubGrp `  G ) )
10 eqid 2296 . . . 4  |-  ( Gs  S )  =  ( Gs  S )
1110subggrp 14640 . . 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 2387 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  S )
1510subgbas 14641 . . . 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 2372 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A  e.  ( Base `  ( Gs  S ) ) )
1816eleq2d 2363 . . . 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 2386 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  S
)  ->  y  e.  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
22 oveq1 5881 . . . . . . 7  |-  ( x  =  n  ->  (
x  .x.  A )  =  ( n  .x.  A ) )
2322cbvmptv 4127 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( n  e.  ZZ  |->  ( n  .x.  A ) )
24 ovex 5899 . . . . . 6  |-  ( n 
.x.  A )  e. 
_V
2523, 24elrnmpti 4946 . . . . 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 14651 . . . . . . 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 2307 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  y  e.  S )  /\  n  e.  ZZ )  ->  (
y  =  ( n 
.x.  A )  <->  y  =  ( n (.g `  ( Gs  S ) ) A ) ) )
3332rexbidva 2573 . . . 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 15190 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( Gs  S )  e. CycGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   ZZcz 10040   Basecbs 13164   ↾s cress 13165   Grpcgrp 14378  .gcmg 14382  SubGrpcsubg 14631  CycGrpccyg 15180
This theorem is referenced by:  cycsubgcyg2  15204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-cyg 15181
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