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Theorem gxinv 20937
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 5866 . . . 4  |-  ( m  =  0  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P 0 ) )
2 oveq2 5866 . . . . 5  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
32fveq2d 5529 . . . 4  |-  ( m  =  0  ->  ( N `  ( A P m ) )  =  ( N `  ( A P 0 ) ) )
41, 3eqeq12d 2297 . . 3  |-  ( m  =  0  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P 0 )  =  ( N `  ( A P 0 ) ) ) )
5 oveq2 5866 . . . 4  |-  ( m  =  k  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P k ) )
6 oveq2 5866 . . . . 5  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
76fveq2d 5529 . . . 4  |-  ( m  =  k  ->  ( N `  ( A P m ) )  =  ( N `  ( A P k ) ) )
85, 7eqeq12d 2297 . . 3  |-  ( m  =  k  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P k )  =  ( N `  ( A P k ) ) ) )
9 oveq2 5866 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P ( k  +  1 ) ) )
10 oveq2 5866 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
1110fveq2d 5529 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( N `  ( A P m ) )  =  ( N `  ( A P ( k  +  1 ) ) ) )
129, 11eqeq12d 2297 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P ( k  +  1 ) )  =  ( N `  ( A P ( k  +  1 ) ) ) ) )
13 oveq2 5866 . . . 4  |-  ( m  =  -u k  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P
-u k ) )
14 oveq2 5866 . . . . 5  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
1514fveq2d 5529 . . . 4  |-  ( m  =  -u k  ->  ( N `  ( A P m ) )  =  ( N `  ( A P -u k
) ) )
1613, 15eqeq12d 2297 . . 3  |-  ( m  =  -u k  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P -u k )  =  ( N `  ( A P -u k ) ) ) )
17 oveq2 5866 . . . 4  |-  ( m  =  K  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P K ) )
18 oveq2 5866 . . . . 5  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
1918fveq2d 5529 . . . 4  |-  ( m  =  K  ->  ( N `  ( A P m ) )  =  ( N `  ( A P K ) ) )
2017, 19eqeq12d 2297 . . 3  |-  ( m  =  K  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P K )  =  ( N `  ( A P K ) ) ) )
21 eqid 2283 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
22 gxinv.2 . . . . . 6  |-  N  =  ( inv `  G
)
2321, 22grpoinvid 20899 . . . . 5  |-  ( G  e.  GrpOp  ->  ( N `  (GId `  G )
)  =  (GId `  G ) )
2423adantr 451 . . . 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 20928 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2827fveq2d 5529 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  ( N `  (GId `  G ) ) )
2925, 22grpoinvcl 20893 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
3025, 21, 26gx0 20928 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
( N `  A
) P 0 )  =  (GId `  G
) )
3129, 30syldan 456 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) P 0 )  =  (GId `  G
) )
3224, 28, 313eqtr4rd 2326 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) P 0 )  =  ( N `  ( A P 0 ) ) )
33293adant3 975 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  A )  e.  X )
3425, 26gxnn0suc 20931 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X  /\  k  e.  NN0 )  ->  (
( N `  A
) P ( k  +  1 ) )  =  ( ( ( N `  A ) P k ) G ( N `  A
) ) )
3533, 34syld3an2 1229 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( N `  A
) P ( k  +  1 ) )  =  ( ( ( N `  A ) P k ) G ( N `  A
) ) )
36 nn0z 10046 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  ZZ )
3725, 26gxcom 20936 . . . . . . . . . 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 ) ) )
3836, 37syl3an3 1217 . . . . . . . . 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 ) ) )
3933, 38syld3an2 1229 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( ( N `  A ) P k ) G ( N `
 A ) )  =  ( ( N `
 A ) G ( ( N `  A ) P k ) ) )
4035, 39eqtrd 2315 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( N `  A
) P ( k  +  1 ) )  =  ( ( N `
 A ) G ( ( N `  A ) P k ) ) )
4140adantr 451 . . . . . 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 ) ) )
4225, 26gxnn0suc 20931 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
4342fveq2d 5529 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  ( A P ( k  +  1 ) ) )  =  ( N `  ( ( A P k ) G A ) ) )
44 simp1 955 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  G  e.  GrpOp )
4525, 26gxcl 20932 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P k )  e.  X )
4636, 45syl3an3 1217 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P k )  e.  X )
47 simp2 956 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  A  e.  X )
4825, 22grpoinvop 20908 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X  /\  A  e.  X )  ->  ( N `  ( ( A P k ) G A ) )  =  ( ( N `  A ) G ( N `  ( A P k ) ) ) )
4944, 46, 47, 48syl3anc 1182 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  ( ( A P k ) G A ) )  =  ( ( N `  A ) G ( N `  ( A P k ) ) ) )
5043, 49eqtrd 2315 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( N `  ( A P ( k  +  1 ) ) )  =  ( ( N `
 A ) G ( N `  ( A P k ) ) ) )
5150adantr 451 . . . . . . 7  |-  ( ( ( 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 ) ) ) )
52 oveq2 5866 . . . . . . . 8  |-  ( ( ( N `  A
) P k )  =  ( N `  ( A P k ) )  ->  ( ( N `  A ) G ( ( N `
 A ) P k ) )  =  ( ( N `  A ) G ( N `  ( A P k ) ) ) )
5352adantl 452 . . . . . . 7  |-  ( ( ( 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 ) ) ) )
5451, 53eqtr4d 2318 . . . . . 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 ) ) )
5541, 54eqtr4d 2318 . . . . 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 ) ) ) )
5655ex 423 . . . 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 ) ) ) ) )
57563expia 1153 . . 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 ) ) ) ) ) )
58 nnz 10045 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
59293adant3 975 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( N `  A )  e.  X )
6025, 22, 26gxneg 20933 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X  /\  k  e.  ZZ )  ->  (
( N `  A
) P -u k
)  =  ( N `
 ( ( N `
 A ) P k ) ) )
6159, 60syld3an2 1229 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  (
( N `  A
) P -u k
)  =  ( N `
 ( ( N `
 A ) P k ) ) )
6261adantr 451 . . . . . . 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 ) ) )
6325, 22, 26gxneg 20933 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  =  ( N `  ( A P k ) ) )
6463adantr 451 . . . . . . . . 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 ) ) )
65 simpr 447 . . . . . . . . 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 ) ) )
6664, 65eqtr4d 2318 . . . . . . . 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 ) )
6766fveq2d 5529 . . . . . . 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 ) ) )
6862, 67eqtr4d 2318 . . . . . 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 ) ) )
6968ex 423 . . . . 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 ) ) ) )
70693expia 1153 . . . 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 ) ) ) ) )
7158, 70syl5 28 . . 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 ) ) ) ) )
724, 8, 12, 16, 20, 32, 57, 71zindd 10113 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( ( N `  A
) P K )  =  ( N `  ( A P K ) ) ) )
73723impia 1148 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   ^gcgx 20857
This theorem is referenced by:  gxinv2  20938  gxmul  20945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gx 20862
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