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Theorem absexp 12064
Description: Absolute value of natural number exponentiation. (Contributed by NM, 5-Jan-2006.)
Assertion
Ref Expression
absexp  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )

Proof of Theorem absexp
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6048 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5691 . . . . 5  |-  ( j  =  0  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ 0 ) ) )
3 oveq2 6048 . . . . 5  |-  ( j  =  0  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ 0 ) )
42, 3eqeq12d 2418 . . . 4  |-  ( j  =  0  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ 0 ) )  =  ( ( abs `  A ) ^ 0 ) ) )
54imbi2d 308 . . 3  |-  ( j  =  0  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ 0 ) )  =  ( ( abs `  A
) ^ 0 ) ) ) )
6 oveq2 6048 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5691 . . . . 5  |-  ( j  =  k  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ k ) ) )
8 oveq2 6048 . . . . 5  |-  ( j  =  k  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ k
) )
97, 8eqeq12d 2418 . . . 4  |-  ( j  =  k  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) ) )
109imbi2d 308 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ k
) )  =  ( ( abs `  A
) ^ k ) ) ) )
11 oveq2 6048 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5691 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 6048 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) )
1412, 13eqeq12d 2418 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ ( k  +  1 ) ) )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) ) )
1514imbi2d 308 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ (
k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) ) ) )
16 oveq2 6048 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5691 . . . . 5  |-  ( j  =  N  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ N ) ) )
18 oveq2 6048 . . . . 5  |-  ( j  =  N  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ N
) )
1917, 18eqeq12d 2418 . . . 4  |-  ( j  =  N  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
2019imbi2d 308 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ N
) )  =  ( ( abs `  A
) ^ N ) ) ) )
21 abs1 12057 . . . 4  |-  ( abs `  1 )  =  1
22 exp0 11341 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2322fveq2d 5691 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( abs `  1
) )
24 abscl 12038 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2524recnd 9070 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
2625exp0d 11472 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 0 )  =  1 )
2721, 23, 263eqtr4a 2462 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( ( abs `  A
) ^ 0 ) )
28 oveq1 6047 . . . . . . . 8  |-  ( ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
)  ->  ( ( abs `  ( A ^
k ) )  x.  ( abs `  A
) )  =  ( ( ( abs `  A
) ^ k )  x.  ( abs `  A
) ) )
2928adantl 453 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )  ->  (
( abs `  ( A ^ k ) )  x.  ( abs `  A
) )  =  ( ( ( abs `  A
) ^ k )  x.  ( abs `  A
) ) )
30 expp1 11343 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3130fveq2d 5691 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( abs `  (
( A ^ k
)  x.  A ) ) )
32 expcl 11354 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
33 simpl 444 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
34 absmul 12054 . . . . . . . . . 10  |-  ( ( ( A ^ k
)  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
( A ^ k
)  x.  A ) )  =  ( ( abs `  ( A ^ k ) )  x.  ( abs `  A
) ) )
3532, 33, 34syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  (
( A ^ k
)  x.  A ) )  =  ( ( abs `  ( A ^ k ) )  x.  ( abs `  A
) ) )
3631, 35eqtrd 2436 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( ( abs `  ( A ^ k
) )  x.  ( abs `  A ) ) )
3736adantr 452 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )  ->  ( abs `  ( A ^
( k  +  1 ) ) )  =  ( ( abs `  ( A ^ k ) )  x.  ( abs `  A
) ) )
38 expp1 11343 . . . . . . . . 9  |-  ( ( ( abs `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ ( k  +  1 ) )  =  ( ( ( abs `  A ) ^ k )  x.  ( abs `  A
) ) )
3925, 38sylan 458 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ ( k  +  1 ) )  =  ( ( ( abs `  A ) ^ k )  x.  ( abs `  A
) ) )
4039adantr 452 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )  ->  (
( abs `  A
) ^ ( k  +  1 ) )  =  ( ( ( abs `  A ) ^ k )  x.  ( abs `  A
) ) )
4129, 37, 403eqtr4d 2446 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )  ->  ( abs `  ( A ^
( k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) )
4241exp31 588 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  ->  ( ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
)  ->  ( abs `  ( A ^ (
k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) ) ) )
4342com12 29 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  CC  ->  (
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
)  ->  ( abs `  ( A ^ (
k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) ) ) )
4443a2d 24 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  CC  ->  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )  ->  ( A  e.  CC  ->  ( abs `  ( A ^ ( k  +  1 ) ) )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) ) ) )
455, 10, 15, 20, 27, 44nn0ind 10322 . 2  |-  ( N  e.  NN0  ->  ( A  e.  CC  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
4645impcom 420 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   NN0cn0 10177   ^cexp 11337   abscabs 11994
This theorem is referenced by:  absexpz  12065  abssq  12066  sqabs  12067  absexpd  12209  expcnv  12598  eftabs  12633  efcllem  12635  efaddlem  12650  iblabsr  19674  iblmulc2  19675  abelthlem7  20307  efif1olem3  20399  efif1olem4  20400  logtayllem  20503  bndatandm  20722  ftalem1  20808  mule1  20884  iblmulc2nc  26169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996
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