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Theorem expnegzap 10214
Description: Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expnegzap |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
Proof of Theorem expnegzap
StepHypRef Expression
1 elznn0 8967 . . 3 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
2 expnegap0 10188 . . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
323expia 1164 . . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> ( N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
43adantr 272 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
5 simpl 108 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A e. CC /\ A # 0 ) )
6 simprl 503 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. RR )
76recnd 7712 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. CC )
8 simprr 504 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. NN0 )
9 expineg2 10189 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
105, 7, 8, 9syl12anc 1195 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
1110oveq2d 5742 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( 1 / ( A ^ -u N ) ) ) )
12 expcl 10198 . . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
1312ad2ant2rl 500 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) e. CC )
14 simpll 501 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A e. CC )
15 simplr 502 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A # 0 )
168nn0zd 9069 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. ZZ )
17 expap0i 10212 . . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
1814, 15, 16, 17syl3anc 1197 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) # 0 )
1913, 18recrecapd 8452 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( 1 / ( A ^ -u N ) ) ) = ( A ^ -u N ) )
2011, 19eqtr2d 2146 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
2120expr 370 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( -u N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
224, 21jaod 689 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( ( N e. NN0 \/ -u N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
2322expimpd 358 . . 3 |- ( ( A e. CC /\ A # 0 ) -> ( ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
241, 23syl5bi 151 . 2 |- ( ( A e. CC /\ A # 0 ) -> ( N e. ZZ -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
25243impia 1159 1 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   \/ wo 680   /\ w3a 943   = wceq 1312   e. wcel 1461   class class class wbr 3893  (class class class)co 5726   CCcc 7539   RRcr 7540   0cc0 7541   1c1 7542   -ucneg 7851   # cap 8255   / cdiv 8339   NN0cn0 8875   ZZcz 8952   ^cexp 10179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-n0 8876  df-z 8953  df-uz 9223  df-seqfrec 10106  df-exp 10180
This theorem is referenced by:  expsubap  10228  expnegapd  10318
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