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Theorem absexp 11105
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 5898 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5533 . . . . 5  |-  ( j  =  0  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ 0 ) ) )
3 oveq2 5898 . . . . 5  |-  ( j  =  0  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ 0 ) )
42, 3eqeq12d 2203 . . . 4  |-  ( j  =  0  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ 0 ) )  =  ( ( abs `  A ) ^ 0 ) ) )
54imbi2d 230 . . 3  |-  ( j  =  0  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ 0 ) )  =  ( ( abs `  A
) ^ 0 ) ) ) )
6 oveq2 5898 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5533 . . . . 5  |-  ( j  =  k  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ k ) ) )
8 oveq2 5898 . . . . 5  |-  ( j  =  k  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ k
) )
97, 8eqeq12d 2203 . . . 4  |-  ( j  =  k  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) ) )
109imbi2d 230 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ k
) )  =  ( ( abs `  A
) ^ k ) ) ) )
11 oveq2 5898 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5533 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 5898 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) )
1412, 13eqeq12d 2203 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ ( k  +  1 ) ) )  =  ( ( abs `  A ) ^ (
k  +  1 ) ) ) )
1514imbi2d 230 . . 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 5898 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5533 . . . . 5  |-  ( j  =  N  ->  ( abs `  ( A ^
j ) )  =  ( abs `  ( A ^ N ) ) )
18 oveq2 5898 . . . . 5  |-  ( j  =  N  ->  (
( abs `  A
) ^ j )  =  ( ( abs `  A ) ^ N
) )
1917, 18eqeq12d 2203 . . . 4  |-  ( j  =  N  ->  (
( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
)  <->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
2019imbi2d 230 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( abs `  ( A ^ j ) )  =  ( ( abs `  A ) ^ j
) )  <->  ( A  e.  CC  ->  ( abs `  ( A ^ N
) )  =  ( ( abs `  A
) ^ N ) ) ) )
21 abs1 11098 . . . 4  |-  ( abs `  1 )  =  1
22 exp0 10541 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2322fveq2d 5533 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( abs `  1
) )
24 abscl 11077 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2524recnd 8003 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
2625exp0d 10665 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 0 )  =  1 )
2721, 23, 263eqtr4a 2247 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( A ^
0 ) )  =  ( ( abs `  A
) ^ 0 ) )
28 oveq1 5897 . . . . . . . 8  |-  ( ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
)  ->  ( ( abs `  ( A ^
k ) )  x.  ( abs `  A
) )  =  ( ( ( abs `  A
) ^ k )  x.  ( abs `  A
) ) )
2928adantl 277 . . . . . . 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 10544 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3130fveq2d 5533 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( abs `  (
( A ^ k
)  x.  A ) ) )
32 expcl 10555 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
33 simpl 109 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
34 absmul 11095 . . . . . . . . . 10  |-  ( ( ( A ^ k
)  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
( A ^ k
)  x.  A ) )  =  ( ( abs `  ( A ^ k ) )  x.  ( abs `  A
) ) )
3532, 33, 34syl2anc 411 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  (
( A ^ k
)  x.  A ) )  =  ( ( abs `  ( A ^ k ) )  x.  ( abs `  A
) ) )
3631, 35eqtrd 2221 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ ( k  +  1 ) ) )  =  ( ( abs `  ( A ^ k
) )  x.  ( abs `  A ) ) )
3736adantr 276 . . . . . . 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 10544 . . . . . . . . 9  |-  ( ( ( abs `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ ( k  +  1 ) )  =  ( ( ( abs `  A ) ^ k )  x.  ( abs `  A
) ) )
3925, 38sylan 283 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ ( k  +  1 ) )  =  ( ( ( abs `  A ) ^ k )  x.  ( abs `  A
) ) )
4039adantr 276 . . . . . . 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 2231 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )  ->  ( abs `  ( A ^
( k  +  1 ) ) )  =  ( ( abs `  A
) ^ ( k  +  1 ) ) )
4241exp31 364 . . . . 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 9384 . 2  |-  ( N  e.  NN0  ->  ( A  e.  CC  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
4645impcom 125 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2159   ` cfv 5230  (class class class)co 5890   CCcc 7826   0cc0 7828   1c1 7829    + caddc 7831    x. cmul 7833   NN0cn0 9193   ^cexp 10536   abscabs 11023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-rp 9671  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025
This theorem is referenced by:  absexpzap  11106  abssq  11107  sqabs  11108  absexpd  11218  expcnvap0  11527  expcnv  11529  eftabs  11681  efaddlem  11699
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