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Theorem gxinv 20953
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 5882 . . . 4  |-  ( m  =  0  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P 0 ) )
2 oveq2 5882 . . . . 5  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
32fveq2d 5545 . . . 4  |-  ( m  =  0  ->  ( N `  ( A P m ) )  =  ( N `  ( A P 0 ) ) )
41, 3eqeq12d 2310 . . 3  |-  ( m  =  0  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P 0 )  =  ( N `  ( A P 0 ) ) ) )
5 oveq2 5882 . . . 4  |-  ( m  =  k  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P k ) )
6 oveq2 5882 . . . . 5  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
76fveq2d 5545 . . . 4  |-  ( m  =  k  ->  ( N `  ( A P m ) )  =  ( N `  ( A P k ) ) )
85, 7eqeq12d 2310 . . 3  |-  ( m  =  k  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P k )  =  ( N `  ( A P k ) ) ) )
9 oveq2 5882 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P ( k  +  1 ) ) )
10 oveq2 5882 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
1110fveq2d 5545 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( N `  ( A P m ) )  =  ( N `  ( A P ( k  +  1 ) ) ) )
129, 11eqeq12d 2310 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P ( k  +  1 ) )  =  ( N `  ( A P ( k  +  1 ) ) ) ) )
13 oveq2 5882 . . . 4  |-  ( m  =  -u k  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P
-u k ) )
14 oveq2 5882 . . . . 5  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
1514fveq2d 5545 . . . 4  |-  ( m  =  -u k  ->  ( N `  ( A P m ) )  =  ( N `  ( A P -u k
) ) )
1613, 15eqeq12d 2310 . . 3  |-  ( m  =  -u k  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P -u k )  =  ( N `  ( A P -u k ) ) ) )
17 oveq2 5882 . . . 4  |-  ( m  =  K  ->  (
( N `  A
) P m )  =  ( ( N `
 A ) P K ) )
18 oveq2 5882 . . . . 5  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
1918fveq2d 5545 . . . 4  |-  ( m  =  K  ->  ( N `  ( A P m ) )  =  ( N `  ( A P K ) ) )
2017, 19eqeq12d 2310 . . 3  |-  ( m  =  K  ->  (
( ( N `  A ) P m )  =  ( N `
 ( A P m ) )  <->  ( ( N `  A ) P K )  =  ( N `  ( A P K ) ) ) )
21 eqid 2296 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
22 gxinv.2 . . . . . 6  |-  N  =  ( inv `  G
)
2321, 22grpoinvid 20915 . . . . 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 20944 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2827fveq2d 5545 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  ( N `  (GId `  G ) ) )
2925, 22grpoinvcl 20909 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
3025, 21, 26gx0 20944 . . . . 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 2339 . . 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 20947 . . . . . . . . 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 10062 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  ZZ )
3725, 26gxcom 20952 . . . . . . . . . 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 2328 . . . . . . 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 20947 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
4342fveq2d 5545 . . . . . . . . 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 20948 . . . . . . . . . . 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 20924 . . . . . . . . . 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 2328 . . . . . . . 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 5882 . . . . . . . 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 2331 . . . . . 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 2331 . . . . 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 10061 . . . 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 20949 . . . . . . . . 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 20949 . . . . . . . . . 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 2331 . . . . . . . 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 5545 . . . . . . 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 2331 . . . . . 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 10129 . 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 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   -ucneg 9054   NNcn 9762   NN0cn0 9981   ZZcz 10040   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871   ^gcgx 20873
This theorem is referenced by:  gxinv2  20954  gxmul  20961
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-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-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-seq 11063  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gx 20878
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