MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  absexp Unicode version

Theorem absexp 12038
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 6030 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5674 . . . . 5  |-  ( j  =  0  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ 0 ) ) )
3 oveq2 6030 . . . . 5  |-  ( j  =  0  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ 0 ) )
42, 3eqeq12d 2403 . . . 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 6030 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5674 . . . . 5  |-  ( j  =  k  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ k ) ) )
8 oveq2 6030 . . . . 5  |-  ( j  =  k  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ k
) )
97, 8eqeq12d 2403 . . . 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 6030 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5674 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 6030 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) )
1412, 13eqeq12d 2403 . . . 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 6030 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5674 . . . . 5  |-  ( j  =  N  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ N ) ) )
18 oveq2 6030 . . . . 5  |-  ( j  =  N  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ N
) )
1917, 18eqeq12d 2403 . . . 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 12031 . . . 4  |-  ( abs `  1 )  =  1
22 exp0 11315 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2322fveq2d 5674 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( abs `  1
) )
24 abscl 12012 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2524recnd 9049 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
2625exp0d 11446 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 0 )  =  1 )
2721, 23, 263eqtr4a 2447 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( ( abs `  A
) ^ 0 ) )
28 oveq1 6029 . . . . . . . 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 11317 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3130fveq2d 5674 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( abs `  (
( A ^ k
)  x.  A ) ) )
32 expcl 11328 . . . . . . . . . 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 12028 . . . . . . . . . 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 2421 . . . . . . . 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 11317 . . . . . . . . 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 2431 . . . . . 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 10300 . 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 1717   ` cfv 5396  (class class class)co 6022   CCcc 8923   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930   NN0cn0 10155   ^cexp 11311   abscabs 11968
This theorem is referenced by:  absexpz  12039  abssq  12040  sqabs  12041  absexpd  12183  expcnv  12572  eftabs  12607  efcllem  12609  efaddlem  12624  iblabsr  19590  iblmulc2  19591  abelthlem7  20223  efif1olem3  20315  efif1olem4  20316  logtayllem  20419  bndatandm  20638  ftalem1  20724  mule1  20800  iblmulc2nc  25972
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970
  Copyright terms: Public domain W3C validator