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Theorem absexp 12101
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 6081 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5724 . . . . 5  |-  ( j  =  0  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ 0 ) ) )
3 oveq2 6081 . . . . 5  |-  ( j  =  0  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ 0 ) )
42, 3eqeq12d 2449 . . . 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 6081 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5724 . . . . 5  |-  ( j  =  k  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ k ) ) )
8 oveq2 6081 . . . . 5  |-  ( j  =  k  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ k
) )
97, 8eqeq12d 2449 . . . 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 6081 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5724 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 6081 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) )
1412, 13eqeq12d 2449 . . . 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 6081 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5724 . . . . 5  |-  ( j  =  N  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ N ) ) )
18 oveq2 6081 . . . . 5  |-  ( j  =  N  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ N
) )
1917, 18eqeq12d 2449 . . . 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 12094 . . . 4  |-  ( abs `  1 )  =  1
22 exp0 11378 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2322fveq2d 5724 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( abs `  1
) )
24 abscl 12075 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2524recnd 9106 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
2625exp0d 11509 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 0 )  =  1 )
2721, 23, 263eqtr4a 2493 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( ( abs `  A
) ^ 0 ) )
28 oveq1 6080 . . . . . . . 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 11380 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3130fveq2d 5724 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( abs `  (
( A ^ k
)  x.  A ) ) )
32 expcl 11391 . . . . . . . . . 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 12091 . . . . . . . . . 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 2467 . . . . . . . 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 11380 . . . . . . . . 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 2477 . . . . . 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 10358 . 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987   NN0cn0 10213   ^cexp 11374   abscabs 12031
This theorem is referenced by:  absexpz  12102  abssq  12103  sqabs  12104  absexpd  12246  expcnv  12635  eftabs  12670  efcllem  12672  efaddlem  12687  iblabsr  19713  iblmulc2  19714  abelthlem7  20346  efif1olem3  20438  efif1olem4  20439  logtayllem  20542  bndatandm  20761  ftalem1  20847  mule1  20923  iblmulc2nc  26260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
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