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Theorem absexp 10737
Description: Absolute value of positive integer 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 5734 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5377 . . . . 5  |-  ( j  =  0  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ 0 ) ) )
3 oveq2 5734 . . . . 5  |-  ( j  =  0  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ 0 ) )
42, 3eqeq12d 2127 . . . 4  |-  ( j  =  0  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ 0 ) )  =  ( ( abs `  A ) ^ 0 ) ) )
54imbi2d 229 . . 3  |-  ( j  =  0  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ 0 ) )  =  ( ( abs `  A
) ^ 0 ) ) ) )
6 oveq2 5734 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5377 . . . . 5  |-  ( j  =  k  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ k ) ) )
8 oveq2 5734 . . . . 5  |-  ( j  =  k  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ k
) )
97, 8eqeq12d 2127 . . . 4  |-  ( j  =  k  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) ) )
109imbi2d 229 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ k
) )  =  ( ( abs `  A
) ^ k ) ) ) )
11 oveq2 5734 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5377 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 5734 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) )
1412, 13eqeq12d 2127 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ ( k  +  1 ) ) )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) ) )
1514imbi2d 229 . . 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 5734 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5377 . . . . 5  |-  ( j  =  N  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ N ) ) )
18 oveq2 5734 . . . . 5  |-  ( j  =  N  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ N
) )
1917, 18eqeq12d 2127 . . . 4  |-  ( j  =  N  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
2019imbi2d 229 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ N
) )  =  ( ( abs `  A
) ^ N ) ) ) )
21 abs1 10730 . . . 4  |-  ( abs `  1 )  =  1
22 exp0 10184 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2322fveq2d 5377 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( abs `  1
) )
24 abscl 10709 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2524recnd 7712 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
2625exp0d 10305 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 0 )  =  1 )
2721, 23, 263eqtr4a 2171 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( ( abs `  A
) ^ 0 ) )
28 oveq1 5733 . . . . . . . 8  |-  ( ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
)  ->  ( ( abs `  ( A ^
k ) )  x.  ( abs `  A
) )  =  ( ( ( abs `  A
) ^ k )  x.  ( abs `  A
) ) )
2928adantl 273 . . . . . . 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 10187 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3130fveq2d 5377 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( abs `  (
( A ^ k
)  x.  A ) ) )
32 expcl 10198 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
33 simpl 108 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
34 absmul 10727 . . . . . . . . . 10  |-  ( ( ( A ^ k
)  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
( A ^ k
)  x.  A ) )  =  ( ( abs `  ( A ^ k ) )  x.  ( abs `  A
) ) )
3532, 33, 34syl2anc 406 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  (
( A ^ k
)  x.  A ) )  =  ( ( abs `  ( A ^ k ) )  x.  ( abs `  A
) ) )
3631, 35eqtrd 2145 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( ( abs `  ( A ^ k
) )  x.  ( abs `  A ) ) )
3736adantr 272 . . . . . . 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 10187 . . . . . . . . 9  |-  ( ( ( abs `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ ( k  +  1 ) )  =  ( ( ( abs `  A ) ^ k )  x.  ( abs `  A
) ) )
3925, 38sylan 279 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ ( k  +  1 ) )  =  ( ( ( abs `  A ) ^ k )  x.  ( abs `  A
) ) )
4039adantr 272 . . . . . . 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 2155 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )  ->  ( abs `  ( A ^
( k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) )
4241exp31 359 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  ->  ( ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
)  ->  ( abs `  ( A ^ (
k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) ) ) )
4342com12 30 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  CC  ->  (
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
)  ->  ( abs `  ( A ^ (
k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) ) ) )
4443a2d 26 . . 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 9063 . 2  |-  ( N  e.  NN0  ->  ( A  e.  CC  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
4645impcom 124 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1312    e. wcel 1461   ` cfv 5079  (class class class)co 5726   CCcc 7539   0cc0 7541   1c1 7542    + caddc 7544    x. cmul 7546   NN0cn0 8875   ^cexp 10179   abscabs 10655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658  ax-caucvg 7659
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-n0 8876  df-z 8953  df-uz 9223  df-rp 9338  df-seqfrec 10106  df-exp 10180  df-cj 10501  df-re 10502  df-im 10503  df-rsqrt 10656  df-abs 10657
This theorem is referenced by:  absexpzap  10738  abssq  10739  sqabs  10740  absexpd  10850  expcnvap0  11157  expcnv  11159  eftabs  11207  efaddlem  11225
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