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Theorem absexpz 12786
Description: Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)
Assertion
Ref Expression
absexpz  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) )

Proof of Theorem absexpz
StepHypRef Expression
1 elznn0nn 10652 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 absexp 12785 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
32ex 434 . . . . 5  |-  ( A  e.  CC  ->  ( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
43adantr 465 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
5 ax-1cn 9332 . . . . . . . . 9  |-  1  e.  CC
65a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
1  e.  CC )
7 simpll 753 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 10578 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
98ad2antll 728 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
107, 9expcld 12000 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  e.  CC )
11 simplr 754 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
12 nnz 10660 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
1312ad2antll 728 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
147, 11, 13expne0d 12006 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  =/=  0 )
15 absdiv 12776 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( A ^ -u N
)  e.  CC  /\  ( A ^ -u N
)  =/=  0 )  ->  ( abs `  (
1  /  ( A ^ -u N ) ) )  =  ( ( abs `  1
)  /  ( abs `  ( A ^ -u N
) ) ) )
166, 10, 14, 15syl3anc 1218 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  (
1  /  ( A ^ -u N ) ) )  =  ( ( abs `  1
)  /  ( abs `  ( A ^ -u N
) ) ) )
17 abs1 12778 . . . . . . . . 9  |-  ( abs `  1 )  =  1
1817oveq1i 6096 . . . . . . . 8  |-  ( ( abs `  1 )  /  ( abs `  ( A ^ -u N ) ) )  =  ( 1  /  ( abs `  ( A ^ -u N
) ) )
19 absexp 12785 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( abs `  ( A ^ -u N ) )  =  ( ( abs `  A ) ^ -u N ) )
207, 9, 19syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ -u N ) )  =  ( ( abs `  A ) ^ -u N ) )
2120oveq2d 6102 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  ( abs `  ( A ^ -u N ) ) )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
2218, 21syl5eq 2482 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( abs `  1
)  /  ( abs `  ( A ^ -u N
) ) )  =  ( 1  /  (
( abs `  A
) ^ -u N
) ) )
2316, 22eqtrd 2470 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  (
1  /  ( A ^ -u N ) ) )  =  ( 1  /  ( ( abs `  A ) ^ -u N ) ) )
24 simprl 755 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
2524recnd 9404 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
26 expneg2 11866 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
277, 25, 9, 26syl3anc 1218 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ N
)  =  ( 1  /  ( A ^ -u N ) ) )
2827fveq2d 5690 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( abs `  (
1  /  ( A ^ -u N ) ) ) )
29 abscl 12759 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
3029ad2antrr 725 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  A
)  e.  RR )
3130recnd 9404 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  A
)  e.  CC )
32 expneg2 11866 . . . . . . 7  |-  ( ( ( abs `  A
)  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  (
( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3331, 25, 9, 32syl3anc 1218 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3423, 28, 333eqtr4d 2480 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
3534ex 434 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
364, 35jaod 380 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( N  e. 
NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
37363impia 1184 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
381, 37syl3an3b 1256 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   -ucneg 9588    / cdiv 9985   NNcn 10314   NN0cn0 10571   ZZcz 10638   ^cexp 11857   abscabs 12715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717
This theorem is referenced by:  iseraltlem3  13153  root1cj  22169  lgseisen  22667
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