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Theorem gxinv 21863
Description: The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxinv.1  |-  X  =  ran  G
gxinv.2  |-  N  =  ( inv `  G
)
gxinv.3  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxinv  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  (
( N `  A
) P K )  =  ( N `  ( A P K ) ) )

Proof of Theorem gxinv
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6092 . . . 4  |-  ( m  =  0  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P 0 ) )
2 oveq2 6092 . . . . 5  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
32fveq2d 5735 . . . 4  |-  ( m  =  0  ->  ( N `  ( A P m ) )  =  ( N `  ( A P 0 ) ) )
41, 3eqeq12d 2452 . . 3  |-  ( m  =  0  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P 0 )  =  ( N `  ( A P 0 ) ) ) )
5 oveq2 6092 . . . 4  |-  ( m  =  k  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P k ) )
6 oveq2 6092 . . . . 5  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
76fveq2d 5735 . . . 4  |-  ( m  =  k  ->  ( N `  ( A P m ) )  =  ( N `  ( A P k ) ) )
85, 7eqeq12d 2452 . . 3  |-  ( m  =  k  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P k )  =  ( N `  ( A P k ) ) ) )
9 oveq2 6092 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P ( k  +  1 ) ) )
10 oveq2 6092 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
1110fveq2d 5735 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( N `  ( A P m ) )  =  ( N `  ( A P ( k  +  1 ) ) ) )
129, 11eqeq12d 2452 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P ( k  +  1 ) )  =  ( N `  ( A P ( k  +  1 ) ) ) ) )
13 oveq2 6092 . . . 4  |-  ( m  =  -u k  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P
-u k ) )
14 oveq2 6092 . . . . 5  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
1514fveq2d 5735 . . . 4  |-  ( m  =  -u k  ->  ( N `  ( A P m ) )  =  ( N `  ( A P -u k
) ) )
1613, 15eqeq12d 2452 . . 3  |-  ( m  =  -u k  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P -u k )  =  ( N `  ( A P -u k ) ) ) )
17 oveq2 6092 . . . 4  |-  ( m  =  K  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P K ) )
18 oveq2 6092 . . . . 5  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
1918fveq2d 5735 . . . 4  |-  ( m  =  K  ->  ( N `  ( A P m ) )  =  ( N `  ( A P K ) ) )
2017, 19eqeq12d 2452 . . 3  |-  ( m  =  K  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P K )  =  ( N `  ( A P K ) ) ) )
21 eqid 2438 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
22 gxinv.2 . . . . . 6  |-  N  =  ( inv `  G
)
2321, 22grpoinvid 21825 . . . . 5  |-  ( G  e.  GrpOp  ->  ( N `  (GId `  G )
)  =  (GId `  G ) )
2423adantr 453 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  (GId `  G
) )  =  (GId
`  G ) )
25 gxinv.1 . . . . . 6  |-  X  =  ran  G
26 gxinv.3 . . . . . 6  |-  P  =  ( ^g `  G
)
2725, 21, 26gx0 21854 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2827fveq2d 5735 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  ( N `  (GId `  G ) ) )
2925, 22grpoinvcl 21819 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
3025, 21, 26gx0 21854 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
( N `  A
) P 0 )  =  (GId `  G
) )
3129, 30syldan 458 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) P 0 )  =  (GId `  G
) )
3224, 28, 313eqtr4rd 2481 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) P 0 )  =  ( N `  ( A P 0 ) ) )
33 oveq2 6092 . . . . . . 7  |-  ( ( ( N `  A
) P k )  =  ( N `  ( A P k ) )  ->  ( ( N `  A ) G ( ( N `
 A ) P k ) )  =  ( ( N `  A ) G ( N `  ( A P k ) ) ) )
3433adantl 454 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( ( N `
 A ) G ( ( N `  A ) P k ) )  =  ( ( N `  A
) G ( N `
 ( A P k ) ) ) )
35293adant3 978 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  A )  e.  X )
3625, 26gxnn0suc 21857 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X  /\  k  e.  NN0 )  ->  (
( N `  A
) P ( k  +  1 ) )  =  ( ( ( N `  A ) P k ) G ( N `  A
) ) )
3735, 36syld3an2 1232 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( N `  A
) P ( k  +  1 ) )  =  ( ( ( N `  A ) P k ) G ( N `  A
) ) )
38 nn0z 10309 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  ZZ )
3925, 26gxcom 21862 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X  /\  k  e.  ZZ )  ->  (
( ( N `  A ) P k ) G ( N `
 A ) )  =  ( ( N `
 A ) G ( ( N `  A ) P k ) ) )
4038, 39syl3an3 1220 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X  /\  k  e.  NN0 )  ->  (
( ( N `  A ) P k ) G ( N `
 A ) )  =  ( ( N `
 A ) G ( ( N `  A ) P k ) ) )
4135, 40syld3an2 1232 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( ( N `  A ) P k ) G ( N `
 A ) )  =  ( ( N `
 A ) G ( ( N `  A ) P k ) ) )
4237, 41eqtrd 2470 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( N `  A
) P ( k  +  1 ) )  =  ( ( N `
 A ) G ( ( N `  A ) P k ) ) )
4342adantr 453 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( ( N `
 A ) P ( k  +  1 ) )  =  ( ( N `  A
) G ( ( N `  A ) P k ) ) )
4425, 26gxnn0suc 21857 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
4544fveq2d 5735 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  ( A P ( k  +  1 ) ) )  =  ( N `  ( ( A P k ) G A ) ) )
46 simp1 958 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  G  e.  GrpOp )
4725, 26gxcl 21858 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P k )  e.  X )
4838, 47syl3an3 1220 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P k )  e.  X )
49 simp2 959 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  A  e.  X )
5025, 22grpoinvop 21834 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X  /\  A  e.  X )  ->  ( N `  ( ( A P k ) G A ) )  =  ( ( N `  A ) G ( N `  ( A P k ) ) ) )
5146, 48, 49, 50syl3anc 1185 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  ( ( A P k ) G A ) )  =  ( ( N `  A ) G ( N `  ( A P k ) ) ) )
5245, 51eqtrd 2470 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  ( A P ( k  +  1 ) ) )  =  ( ( N `
 A ) G ( N `  ( A P k ) ) ) )
5352adantr 453 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( N `  ( A P ( k  +  1 ) ) )  =  ( ( N `  A ) G ( N `  ( A P k ) ) ) )
5434, 43, 533eqtr4d 2480 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( ( N `
 A ) P ( k  +  1 ) )  =  ( N `  ( A P ( k  +  1 ) ) ) )
5554ex 425 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( ( N `  A ) P k )  =  ( N `
 ( A P k ) )  -> 
( ( N `  A ) P ( k  +  1 ) )  =  ( N `
 ( A P ( k  +  1 ) ) ) ) )
56553expia 1156 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN0  ->  ( ( ( N `  A ) P k )  =  ( N `
 ( A P k ) )  -> 
( ( N `  A ) P ( k  +  1 ) )  =  ( N `
 ( A P ( k  +  1 ) ) ) ) ) )
57 nnz 10308 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
58293adant3 978 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( N `  A )  e.  X )
5925, 22, 26gxneg 21859 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X  /\  k  e.  ZZ )  ->  (
( N `  A
) P -u k
)  =  ( N `
 ( ( N `
 A ) P k ) ) )
6058, 59syld3an2 1232 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  (
( N `  A
) P -u k
)  =  ( N `
 ( ( N `
 A ) P k ) ) )
6160adantr 453 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( ( N `
 A ) P
-u k )  =  ( N `  (
( N `  A
) P k ) ) )
6225, 22, 26gxneg 21859 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  =  ( N `  ( A P k ) ) )
6362adantr 453 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( A P
-u k )  =  ( N `  ( A P k ) ) )
64 simpr 449 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( ( N `
 A ) P k )  =  ( N `  ( A P k ) ) )
6563, 64eqtr4d 2473 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( A P
-u k )  =  ( ( N `  A ) P k ) )
6665fveq2d 5735 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( N `  ( A P -u k
) )  =  ( N `  ( ( N `  A ) P k ) ) )
6761, 66eqtr4d 2473 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( N `  A ) P k )  =  ( N `
 ( A P k ) ) )  ->  ( ( N `
 A ) P
-u k )  =  ( N `  ( A P -u k ) ) )
6867ex 425 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  (
( ( N `  A ) P k )  =  ( N `
 ( A P k ) )  -> 
( ( N `  A ) P -u k )  =  ( N `  ( A P -u k ) ) ) )
69683expia 1156 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  ZZ  ->  ( ( ( N `  A ) P k )  =  ( N `
 ( A P k ) )  -> 
( ( N `  A ) P -u k )  =  ( N `  ( A P -u k ) ) ) ) )
7057, 69syl5 31 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN  ->  ( ( ( N `  A ) P k )  =  ( N `
 ( A P k ) )  -> 
( ( N `  A ) P -u k )  =  ( N `  ( A P -u k ) ) ) ) )
714, 8, 12, 16, 20, 32, 56, 70zindd 10376 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( ( N `  A
) P K )  =  ( N `  ( A P K ) ) ) )
72713impia 1151 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  (
( N `  A
) P K )  =  ( N `  ( A P K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4882   ` cfv 5457  (class class class)co 6084   0cc0 8995   1c1 8996    + caddc 8998   -ucneg 9297   NNcn 10005   NN0cn0 10226   ZZcz 10287   GrpOpcgr 21779  GIdcgi 21780   invcgn 21781   ^gcgx 21783
This theorem is referenced by:  gxinv2  21864  gxmul  21871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gx 21788
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