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Theorem absexpzap 9650
Description: Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
Assertion
Ref Expression
absexpzap  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) )

Proof of Theorem absexpzap
StepHypRef Expression
1 elznn0nn 8257 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 absexp 9649 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
32ex 108 . . . . 5  |-  ( A  e.  CC  ->  ( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
43adantr 261 . . . 4  |-  ( ( A  e.  CC  /\  A #  0 )  ->  ( N  e.  NN0  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
5 1cnd 7041 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  1  e.  CC )
6 simpll 481 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
7 nnnn0 8186 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
87ad2antll 460 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
96, 8expcld 9355 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u N )  e.  CC )
10 simplr 482 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
11 nnz 8262 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
1211ad2antll 460 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
136, 10, 12expap0d 9361 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u N ) #  0 )
14 absdivap 9642 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( A ^ -u N
)  e.  CC  /\  ( A ^ -u N
) #  0 )  -> 
( abs `  (
1  /  ( A ^ -u N ) ) )  =  ( ( abs `  1
)  /  ( abs `  ( A ^ -u N
) ) ) )
155, 9, 13, 14syl3anc 1135 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  ( 1  / 
( A ^ -u N
) ) )  =  ( ( abs `  1
)  /  ( abs `  ( A ^ -u N
) ) ) )
16 abs1 9644 . . . . . . . . 9  |-  ( abs `  1 )  =  1
1716oveq1i 5522 . . . . . . . 8  |-  ( ( abs `  1 )  /  ( abs `  ( A ^ -u N ) ) )  =  ( 1  /  ( abs `  ( A ^ -u N
) ) )
18 absexp 9649 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( abs `  ( A ^ -u N ) )  =  ( ( abs `  A ) ^ -u N ) )
196, 8, 18syl2anc 391 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  ( A ^ -u N ) )  =  ( ( abs `  A
) ^ -u N
) )
2019oveq2d 5528 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
1  /  ( abs `  ( A ^ -u N
) ) )  =  ( 1  /  (
( abs `  A
) ^ -u N
) ) )
2117, 20syl5eq 2084 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( abs `  1
)  /  ( abs `  ( A ^ -u N
) ) )  =  ( 1  /  (
( abs `  A
) ^ -u N
) ) )
2215, 21eqtrd 2072 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  ( 1  / 
( A ^ -u N
) ) )  =  ( 1  /  (
( abs `  A
) ^ -u N
) ) )
23 simprl 483 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
2423recnd 7052 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
25 expineg2 9238 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
266, 10, 24, 8, 25syl22anc 1136 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
2726fveq2d 5182 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  ( A ^ N ) )  =  ( abs `  (
1  /  ( A ^ -u N ) ) ) )
28 abscl 9623 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2928ad2antrr 457 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  A )  e.  RR )
3029recnd 7052 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  A )  e.  CC )
31 abs00ap 9634 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( abs `  A
) #  0  <->  A #  0
) )
3231ad2antrr 457 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( abs `  A
) #  0  <->  A #  0
) )
3310, 32mpbird 156 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  A ) #  0 )
34 expineg2 9238 . . . . . . 7  |-  ( ( ( ( abs `  A
)  e.  CC  /\  ( abs `  A ) #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  -> 
( ( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3530, 33, 24, 8, 34syl22anc 1136 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( abs `  A
) ^ N )  =  ( 1  / 
( ( abs `  A
) ^ -u N
) ) )
3622, 27, 353eqtr4d 2082 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) )
3736ex 108 . . . 4  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
( N  e.  RR  /\  -u N  e.  NN )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) ) )
384, 37jaod 637 . . 3  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A
) ^ N ) ) )
39383impia 1101 . 2  |-  ( ( A  e.  CC  /\  A #  0  /\  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N
) )
401, 39syl3an3b 1173 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    /\ wa 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   CCcc 6885   RRcr 6886   0cc0 6887   1c1 6888   -ucneg 7181   # cap 7570    / cdiv 7649   NNcn 7912   NN0cn0 8179   ZZcz 8243   ^cexp 9228   abscabs 9569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6973  ax-resscn 6974  ax-1cn 6975  ax-1re 6976  ax-icn 6977  ax-addcl 6978  ax-addrcl 6979  ax-mulcl 6980  ax-mulrcl 6981  ax-addcom 6982  ax-mulcom 6983  ax-addass 6984  ax-mulass 6985  ax-distr 6986  ax-i2m1 6987  ax-1rid 6989  ax-0id 6990  ax-rnegex 6991  ax-precex 6992  ax-cnre 6993  ax-pre-ltirr 6994  ax-pre-ltwlin 6995  ax-pre-lttrn 6996  ax-pre-apti 6997  ax-pre-ltadd 6998  ax-pre-mulgt0 6999  ax-pre-mulext 7000  ax-arch 7001  ax-caucvg 7002
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813  df-0r 6814  df-1r 6815  df-0 6894  df-1 6895  df-r 6897  df-lt 6900  df-pnf 7060  df-mnf 7061  df-xr 7062  df-ltxr 7063  df-le 7064  df-sub 7182  df-neg 7183  df-reap 7564  df-ap 7571  df-div 7650  df-inn 7913  df-2 7971  df-3 7972  df-4 7973  df-n0 8180  df-z 8244  df-uz 8472  df-rp 8582  df-iseq 9186  df-iexp 9229  df-cj 9416  df-re 9417  df-im 9418  df-rsqrt 9570  df-abs 9571
This theorem is referenced by: (None)
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