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Theorem expnegzap 10327
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 9069 . . 3 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
2 expnegap0 10301 . . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
323expia 1183 . . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> ( N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
43adantr 274 . . . . 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 520 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. RR )
76recnd 7794 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. CC )
8 simprr 521 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. NN0 )
9 expineg2 10302 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
105, 7, 8, 9syl12anc 1214 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
1110oveq2d 5790 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( 1 / ( A ^ -u N ) ) ) )
12 expcl 10311 . . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
1312ad2ant2rl 502 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) e. CC )
14 simpll 518 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A e. CC )
15 simplr 519 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A # 0 )
168nn0zd 9171 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. ZZ )
17 expap0i 10325 . . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
1814, 15, 16, 17syl3anc 1216 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) # 0 )
1913, 18recrecapd 8545 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( 1 / ( A ^ -u N ) ) ) = ( A ^ -u N ) )
2011, 19eqtr2d 2173 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
2120expr 372 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( -u N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
224, 21jaod 706 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( ( N e. NN0 \/ -u N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
2322expimpd 360 . . 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 1178 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 697   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   -ucneg 7934   # cap 8343   / cdiv 8432   NN0cn0 8977   ZZcz 9054   ^cexp 10292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219  df-exp 10293
This theorem is referenced by:  expsubap  10341  expnegapd  10431
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