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Theorem cyggeninv 15186
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 15184 . . . 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 14543 . . 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 5881 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
1110eqeq2d 2307 . . . . . 6  |-  ( n  =  m  ->  (
y  =  ( n 
.x.  X )  <->  y  =  ( m  .x.  X ) ) )
1211cbvrexv 2778 . . . . 5  |-  ( E. n  e.  ZZ  y  =  ( n  .x.  X )  <->  E. m  e.  ZZ  y  =  ( m  .x.  X ) )
13 znegcl 10071 . . . . . . . . 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 10134 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  CC )
1716negnegd 9164 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u -u m  =  m )
1817oveq1d 5889 . . . . . . . . 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 14610 . . . . . . . . . 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 2330 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  (
m  .x.  X )  =  ( -u m  .x.  ( N `  X
) ) )
24 oveq1 5881 . . . . . . . . . 10  |-  ( n  =  -u m  ->  (
n  .x.  ( N `  X ) )  =  ( -u m  .x.  ( N `  X ) ) )
2524eqeq2d 2307 . . . . . . . . 9  |-  ( n  =  -u m  ->  (
( m  .x.  X
)  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  (
-u m  .x.  ( N `  X )
) ) )
2625rspcev 2897 . . . . . . . 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 2302 . . . . . . . 8  |-  ( y  =  ( m  .x.  X )  ->  (
y  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  ( n  .x.  ( N `
 X ) ) ) )
2928rexbidv 2577 . . . . . . 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 2680 . . . . 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 2634 . . 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 15184 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   -ucneg 9054   ZZcz 10040   Basecbs 13164   Grpcgrp 14378   inv gcminusg 14379  .gcmg 14382
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-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508
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