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Theorem cyggeninv 16670
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  =  ( invg `  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 16668 . . . 4  |-  ( G  e.  Grp  ->  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  X ) ) ) )
54simprbda 623 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  X  e.  B )
6 cyggeninv.n . . . 4  |-  N  =  ( invg `  G )
71, 6grpinvcl 15889 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
85, 7syldan 470 . 2  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( N `  X
)  e.  B )
94simplbda 624 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  X ) )
10 oveq1 6282 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
1110eqeq2d 2474 . . . . . 6  |-  ( n  =  m  ->  (
y  =  ( n 
.x.  X )  <->  y  =  ( m  .x.  X ) ) )
1211cbvrexv 3082 . . . . 5  |-  ( E. n  e.  ZZ  y  =  ( n  .x.  X )  <->  E. m  e.  ZZ  y  =  ( m  .x.  X ) )
13 znegcl 10887 . . . . . . . . 9  |-  ( m  e.  ZZ  ->  -u m  e.  ZZ )
1413adantl 466 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u m  e.  ZZ )
15 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  ZZ )
1615zcnd 10956 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  CC )
1716negnegd 9910 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u -u m  =  m )
1817oveq1d 6290 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  ( -u -u m  .x.  X )  =  ( m  .x.  X ) )
19 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  G  e.  Grp )
205ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  X  e.  B )
211, 2, 6mulgneg2 15962 . . . . . . . . . 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 1223 . . . . . . . . 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 2503 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  (
m  .x.  X )  =  ( -u m  .x.  ( N `  X
) ) )
24 oveq1 6282 . . . . . . . . . 10  |-  ( n  =  -u m  ->  (
n  .x.  ( N `  X ) )  =  ( -u m  .x.  ( N `  X ) ) )
2524eqeq2d 2474 . . . . . . . . 9  |-  ( n  =  -u m  ->  (
( m  .x.  X
)  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  (
-u m  .x.  ( N `  X )
) ) )
2625rspcev 3207 . . . . . . . 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 661 . . . . . . 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 2464 . . . . . . . 8  |-  ( y  =  ( m  .x.  X )  ->  (
y  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  ( n  .x.  ( N `
 X ) ) ) )
2928rexbidv 2966 . . . . . . 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 222 . . . . . 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 2948 . . . . 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 217 . . . 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 2865 . . 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 16668 . . 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 465 . 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 915 1  |-  ( ( G  e.  Grp  /\  X  e.  E )  ->  ( N `  X
)  e.  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   {crab 2811    |-> cmpt 4498   ran crn 4993   ` cfv 5579  (class class class)co 6275   -ucneg 9795   ZZcz 10853   Basecbs 14479   Grpcgrp 15716   invgcminusg 15717  .gcmg 15720
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-seq 12064  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-mulg 15854
This theorem is referenced by: (None)
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