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Theorem cyggeninv 15170
Description: The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1  |-  B  =  ( Base `  G
)
iscyg.2  |-  .x.  =  (.g
`  G )
iscyg3.e  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
cyggeninv.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
cyggeninv  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( N `  X
)  e.  E )
Distinct variable groups:    x, n, B    n, N, x    n, X, x    n, G, x    .x. , n, x
Allowed substitution hints:    E( x, n)

Proof of Theorem cyggeninv
Dummy variables  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscyg.1 . . . . 5  |-  B  =  ( Base `  G
)
2 iscyg.2 . . . . 5  |-  .x.  =  (.g
`  G )
3 iscyg3.e . . . . 5  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
41, 2, 3iscyggen2 15168 . . . 4  |-  ( G  e.  Grp  ->  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  X ) ) ) )
54simprbda 606 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  X  e.  B )
6 cyggeninv.n . . . 4  |-  N  =  ( inv g `  G )
71, 6grpinvcl 14527 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
85, 7syldan 456 . 2  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( N `  X
)  e.  B )
94simplbda 607 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  X ) )
10 oveq1 5865 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
1110eqeq2d 2294 . . . . . 6  |-  ( n  =  m  ->  (
y  =  ( n 
.x.  X )  <->  y  =  ( m  .x.  X ) ) )
1211cbvrexv 2765 . . . . 5  |-  ( E. n  e.  ZZ  y  =  ( n  .x.  X )  <->  E. m  e.  ZZ  y  =  ( m  .x.  X ) )
13 znegcl 10055 . . . . . . . . 9  |-  ( m  e.  ZZ  ->  -u m  e.  ZZ )
1413adantl 452 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u m  e.  ZZ )
15 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  ZZ )
1615zcnd 10118 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  CC )
1716negnegd 9148 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u -u m  =  m )
1817oveq1d 5873 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  ( -u -u m  .x.  X )  =  ( m  .x.  X ) )
19 simplll 734 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  G  e.  Grp )
205ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  X  e.  B )
211, 2, 6mulgneg2 14594 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  -u m  e.  ZZ  /\  X  e.  B )  ->  ( -u -u m  .x.  X )  =  (
-u m  .x.  ( N `  X )
) )
2219, 14, 20, 21syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  ( -u -u m  .x.  X )  =  ( -u m  .x.  ( N `  X
) ) )
2318, 22eqtr3d 2317 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  (
m  .x.  X )  =  ( -u m  .x.  ( N `  X
) ) )
24 oveq1 5865 . . . . . . . . . 10  |-  ( n  =  -u m  ->  (
n  .x.  ( N `  X ) )  =  ( -u m  .x.  ( N `  X ) ) )
2524eqeq2d 2294 . . . . . . . . 9  |-  ( n  =  -u m  ->  (
( m  .x.  X
)  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  (
-u m  .x.  ( N `  X )
) ) )
2625rspcev 2884 . . . . . . . 8  |-  ( (
-u m  e.  ZZ  /\  ( m  .x.  X
)  =  ( -u m  .x.  ( N `  X ) ) )  ->  E. n  e.  ZZ  ( m  .x.  X )  =  ( n  .x.  ( N `  X ) ) )
2714, 23, 26syl2anc 642 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  E. n  e.  ZZ  ( m  .x.  X )  =  ( n  .x.  ( N `
 X ) ) )
28 eqeq1 2289 . . . . . . . 8  |-  ( y  =  ( m  .x.  X )  ->  (
y  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  ( n  .x.  ( N `
 X ) ) ) )
2928rexbidv 2564 . . . . . . 7  |-  ( y  =  ( m  .x.  X )  ->  ( E. n  e.  ZZ  y  =  ( n  .x.  ( N `  X
) )  <->  E. n  e.  ZZ  ( m  .x.  X )  =  ( n  .x.  ( N `
 X ) ) ) )
3027, 29syl5ibrcom 213 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  (
y  =  ( m 
.x.  X )  ->  E. n  e.  ZZ  y  =  ( n  .x.  ( N `  X
) ) ) )
3130rexlimdva 2667 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  E )  /\  y  e.  B
)  ->  ( E. m  e.  ZZ  y  =  ( m  .x.  X )  ->  E. n  e.  ZZ  y  =  ( n  .x.  ( N `
 X ) ) ) )
3212, 31syl5bi 208 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  E )  /\  y  e.  B
)  ->  ( E. n  e.  ZZ  y  =  ( n  .x.  X )  ->  E. n  e.  ZZ  y  =  ( n  .x.  ( N `
 X ) ) ) )
3332ralimdva 2621 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  X )  ->  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  ( N `  X
) ) ) )
349, 33mpd 14 . 2  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  ( N `  X
) ) )
351, 2, 3iscyggen2 15168 . . 3  |-  ( G  e.  Grp  ->  (
( N `  X
)  e.  E  <->  ( ( N `  X )  e.  B  /\  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  ( N `
 X ) ) ) ) )
3635adantr 451 . 2  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( ( N `  X )  e.  E  <->  ( ( N `  X
)  e.  B  /\  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  ( N `  X ) ) ) ) )
378, 34, 36mpbir2and 888 1  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( N `  X
)  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   -ucneg 9038   ZZcz 10024   Basecbs 13148   Grpcgrp 14362   inv gcminusg 14363  .gcmg 14366
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-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492
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