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Theorem absexpz 11806
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 10053 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 absexp 11805 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
32ex 423 . . . . 5  |-  ( A  e.  CC  ->  ( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
43adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
5 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
65a1i 10 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
1  e.  CC )
7 simpll 730 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 9988 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
98ad2antll 709 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
107, 9expcld 11261 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  e.  CC )
11 simplr 731 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
12 nnz 10061 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
1312ad2antll 709 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
147, 11, 13expne0d 11267 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  =/=  0 )
15 absdiv 11796 . . . . . . . 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 1182 . . . . . . 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 11798 . . . . . . . . 9  |-  ( abs `  1 )  =  1
1817oveq1i 5884 . . . . . . . 8  |-  ( ( abs `  1 )  /  ( abs `  ( A ^ -u N ) ) )  =  ( 1  /  ( abs `  ( A ^ -u N
) ) )
19 absexp 11805 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( abs `  ( A ^ -u N ) )  =  ( ( abs `  A ) ^ -u N ) )
207, 9, 19syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ -u N ) )  =  ( ( abs `  A ) ^ -u N ) )
2120oveq2d 5890 . . . . . . . 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 2340 . . . . . . 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 2328 . . . . . 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 732 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
2524recnd 8877 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
26 expneg2 11128 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
277, 25, 9, 26syl3anc 1182 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ N
)  =  ( 1  /  ( A ^ -u N ) ) )
2827fveq2d 5545 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( abs `  (
1  /  ( A ^ -u N ) ) ) )
29 abscl 11779 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
3029ad2antrr 706 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  A
)  e.  RR )
3130recnd 8877 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  A
)  e.  CC )
32 expneg2 11128 . . . . . . 7  |-  ( ( ( abs `  A
)  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  (
( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3331, 25, 9, 32syl3anc 1182 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3423, 28, 333eqtr4d 2338 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
3534ex 423 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
364, 35jaod 369 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( N  e. 
NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
37363impia 1148 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
381, 37syl3an3b 1220 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 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   abscabs 11735
This theorem is referenced by:  iseraltlem3  12172  root1cj  20112  lgseisen  20608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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