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Theorem absexpz 12110
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 10295 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 absexp 12109 . . . . . 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 9048 . . . . . . . . 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 10228 . . . . . . . . . 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 11523 . . . . . . . 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 10303 . . . . . . . . . 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 11529 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  =/=  0 )
15 absdiv 12100 . . . . . . . 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 12102 . . . . . . . . 9  |-  ( abs `  1 )  =  1
1817oveq1i 6091 . . . . . . . 8  |-  ( ( abs `  1 )  /  ( abs `  ( A ^ -u N ) ) )  =  ( 1  /  ( abs `  ( A ^ -u N
) ) )
19 absexp 12109 . . . . . . . . . 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 6097 . . . . . . . 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 2480 . . . . . . 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 2468 . . . . . 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 9114 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
26 expneg2 11390 . . . . . . . 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 5732 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( abs `  (
1  /  ( A ^ -u N ) ) ) )
29 abscl 12083 . . . . . . . . 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 9114 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  A
)  e.  CC )
32 expneg2 11390 . . . . . . 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 2478 . . . . 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 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991   -ucneg 9292    / cdiv 9677   NNcn 10000   NN0cn0 10221   ZZcz 10282   ^cexp 11382   abscabs 12039
This theorem is referenced by:  iseraltlem3  12477  root1cj  20640  lgseisen  21137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041
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