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Theorem cyggeninv 16381
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 16379 . . . 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 15604 . . 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 6119 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
1110eqeq2d 2454 . . . . . 6  |-  ( n  =  m  ->  (
y  =  ( n 
.x.  X )  <->  y  =  ( m  .x.  X ) ) )
1211cbvrexv 2969 . . . . 5  |-  ( E. n  e.  ZZ  y  =  ( n  .x.  X )  <->  E. m  e.  ZZ  y  =  ( m  .x.  X ) )
13 znegcl 10701 . . . . . . . . 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 10769 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  m  e.  CC )
1716negnegd 9731 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  -u -u m  =  m )
1817oveq1d 6127 . . . . . . . . 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 15675 . . . . . . . . . 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 1218 . . . . . . . . 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 2477 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  E )  /\  y  e.  B )  /\  m  e.  ZZ )  ->  (
m  .x.  X )  =  ( -u m  .x.  ( N `  X
) ) )
24 oveq1 6119 . . . . . . . . . 10  |-  ( n  =  -u m  ->  (
n  .x.  ( N `  X ) )  =  ( -u m  .x.  ( N `  X ) ) )
2524eqeq2d 2454 . . . . . . . . 9  |-  ( n  =  -u m  ->  (
( m  .x.  X
)  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  (
-u m  .x.  ( N `  X )
) ) )
2625rspcev 3094 . . . . . . . 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 2449 . . . . . . . 8  |-  ( y  =  ( m  .x.  X )  ->  (
y  =  ( n 
.x.  ( N `  X ) )  <->  ( m  .x.  X )  =  ( n  .x.  ( N `
 X ) ) ) )
2928rexbidv 2757 . . . . . . 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 2862 . . . . 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 2815 . . 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 16379 . . 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 913 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 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740    e. cmpt 4371   ran crn 4862   ` cfv 5439  (class class class)co 6112   -ucneg 9617   ZZcz 10667   Basecbs 14195   Grpcgrp 15431   invgcminusg 15432  .gcmg 15435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-seq 11828  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-mulg 15569
This theorem is referenced by: (None)
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