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Theorem absexpz 12065
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 10251 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 absexp 12064 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
32ex 424 . . . . 5  |-  ( A  e.  CC  ->  ( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
43adantr 452 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
5 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
65a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
1  e.  CC )
7 simpll 731 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 10184 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
98ad2antll 710 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
107, 9expcld 11478 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  e.  CC )
11 simplr 732 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
12 nnz 10259 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
1312ad2antll 710 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
147, 11, 13expne0d 11484 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  =/=  0 )
15 absdiv 12055 . . . . . . . 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 1184 . . . . . . 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 12057 . . . . . . . . 9  |-  ( abs `  1 )  =  1
1817oveq1i 6050 . . . . . . . 8  |-  ( ( abs `  1 )  /  ( abs `  ( A ^ -u N ) ) )  =  ( 1  /  ( abs `  ( A ^ -u N
) ) )
19 absexp 12064 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( abs `  ( A ^ -u N ) )  =  ( ( abs `  A ) ^ -u N ) )
207, 9, 19syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ -u N ) )  =  ( ( abs `  A ) ^ -u N ) )
2120oveq2d 6056 . . . . . . . 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 2448 . . . . . . 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 2436 . . . . . 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 733 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
2524recnd 9070 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
26 expneg2 11345 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
277, 25, 9, 26syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ N
)  =  ( 1  /  ( A ^ -u N ) ) )
2827fveq2d 5691 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( abs `  (
1  /  ( A ^ -u N ) ) ) )
29 abscl 12038 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
3029ad2antrr 707 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  A
)  e.  RR )
3130recnd 9070 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  A
)  e.  CC )
32 expneg2 11345 . . . . . . 7  |-  ( ( ( abs `  A
)  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  (
( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3331, 25, 9, 32syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3423, 28, 333eqtr4d 2446 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
3534ex 424 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
364, 35jaod 370 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( N  e. 
NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
37363impia 1150 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
381, 37syl3an3b 1222 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   -ucneg 9248    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ^cexp 11337   abscabs 11994
This theorem is referenced by:  iseraltlem3  12432  root1cj  20593  lgseisen  21090
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996
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