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Theorem expnegzap 10898
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 9555 . . 3 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
2 expnegap0 10872 . . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
323expia 1232 . . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> ( N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
43adantr 276 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
5 simpl 109 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A e. CC /\ A # 0 ) )
6 simprl 531 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. RR )
76recnd 8267 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. CC )
8 simprr 533 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. NN0 )
9 expineg2 10873 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
105, 7, 8, 9syl12anc 1272 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
1110oveq2d 6044 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( 1 / ( A ^ -u N ) ) ) )
12 expcl 10882 . . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
1312ad2ant2rl 511 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) e. CC )
14 simpll 527 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A e. CC )
15 simplr 529 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A # 0 )
168nn0zd 9661 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. ZZ )
17 expap0i 10896 . . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
1814, 15, 16, 17syl3anc 1274 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) # 0 )
1913, 18recrecapd 9024 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( 1 / ( A ^ -u N ) ) ) = ( A ^ -u N ) )
2011, 19eqtr2d 2265 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
2120expr 375 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( -u N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
224, 21jaod 725 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( ( N e. NN0 \/ -u N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
2322expimpd 363 . . 3 |- ( ( A e. CC /\ A # 0 ) -> ( ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
241, 23biimtrid 152 . 2 |- ( ( A e. CC /\ A # 0 ) -> ( N e. ZZ -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
25243impia 1227 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 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   -ucneg 8410   # cap 8820   / cdiv 8911   NN0cn0 9461   ZZcz 9540   ^cexp 10863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773  df-exp 10864
This theorem is referenced by:  expsubap  10912  expnegapd  11005
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