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Theorem cyggeninv 15486
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 15484 . . . 4  |-  ( G  e.  Grp  ->  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  X ) ) ) )
54simprbda 607 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  X  e.  B )
6 cyggeninv.n . . . 4  |-  N  =  ( inv g `  G )
71, 6grpinvcl 14843 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
85, 7syldan 457 . 2  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( N `  X
)  e.  B )
94simplbda 608 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  X ) )
10 oveq1 6081 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
1110eqeq2d 2447 . . . . . 6  |-  ( n  =  m  ->  (
y  =  ( n 
.x.  X )  <->  y  =  ( m  .x.  X ) ) )
1211cbvrexv 2926 . . . . 5  |-  ( E. n  e.  ZZ  y  =  ( n  .x.  X )  <->  E. m  e.  ZZ  y  =  ( m  .x.  X ) )
13 znegcl 10306 . . . . . . . . 9  |-  ( m  e.  ZZ  ->  -u m  e.  ZZ )
1413adantl 453 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u m  e.  ZZ )
15 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  ZZ )
1615zcnd 10369 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  CC )
1716negnegd 9395 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u -u m  =  m )
1817oveq1d 6089 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  ( -u -u m  .x.  X )  =  ( m  .x.  X ) )
19 simplll 735 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  G  e.  Grp )
205ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  X  e.  B )
211, 2, 6mulgneg2 14910 . . . . . . . . . 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 1184 . . . . . . . . 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 2470 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  (
m  .x.  X )  =  ( -u m  .x.  ( N `  X
) ) )
24 oveq1 6081 . . . . . . . . . 10  |-  ( n  =  -u m  ->  (
n  .x.  ( N `  X ) )  =  ( -u m  .x.  ( N `  X ) ) )
2524eqeq2d 2447 . . . . . . . . 9  |-  ( n  =  -u m  ->  (
( m  .x.  X
)  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  (
-u m  .x.  ( N `  X )
) ) )
2625rspcev 3045 . . . . . . . 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 643 . . . . . . 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 2442 . . . . . . . 8  |-  ( y  =  ( m  .x.  X )  ->  (
y  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  ( n  .x.  ( N `
 X ) ) ) )
2928rexbidv 2719 . . . . . . 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 214 . . . . . 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 2823 . . . . 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 209 . . . 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 2777 . . 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 15 . 2  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  ( N `  X
) ) )
351, 2, 3iscyggen2 15484 . . 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 452 . 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 889 1  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( N `  X
)  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2698   E.wrex 2699   {crab 2702    e. cmpt 4259   ran crn 4872   ` cfv 5447  (class class class)co 6074   -ucneg 9285   ZZcz 10275   Basecbs 13462   Grpcgrp 14678   inv gcminusg 14679  .gcmg 14682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-n0 10215  df-z 10276  df-uz 10482  df-fz 11037  df-seq 11317  df-0g 13720  df-mnd 14683  df-grp 14805  df-minusg 14806  df-mulg 14808
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