Theorem expnegzap 9289

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

Step	Hyp	Ref	Expression
1		elznn0 8260	. . 3 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
2		expnegap0 9263	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
3	2	3expia 1106	. . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> ( N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
4	3	adantr 261	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
5		simpl 102	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A e. CC /\ A # 0 ) )
6		simprl 483	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. RR )
7	6	recnd 7054	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> N e. CC )
8		simprr 484	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. NN0 )
9		expineg2 9264	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
10	5, 7, 8, 9	syl12anc 1133	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
11	10	oveq2d 5528	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( 1 / ( A ^ -u N ) ) ) )
12		expcl 9273	. . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
13	12	ad2ant2rl 480	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) e. CC )
14		simpll 481	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A e. CC )
15		simplr 482	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> A # 0 )
16	8	nn0zd 8358	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> -u N e. ZZ )
17		expap0i 9287	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
18	14, 15, 16, 17	syl3anc 1135	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) # 0 )
19	13, 18	recrecapd 7761	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( 1 / ( 1 / ( A ^ -u N ) ) ) = ( A ^ -u N ) )
20	11, 19	eqtr2d 2073	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
21	20	expr 357	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( -u N e. NN0 -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
22	4, 21	jaod 637	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. RR ) -> ( ( N e. NN0 \/ -u N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
23	22	expimpd 345	. . 3 |- ( ( A e. CC /\ A # 0 ) -> ( ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
24	1, 23	syl5bi 141	. 2 |- ( ( A e. CC /\ A # 0 ) -> ( N e. ZZ -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) )
25	24	3impia 1101	1 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   class class class wbr 3764  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   1c1 6890   -ucneg 7183   # cap 7572   / cdiv 7651   NN0cn0 8181   ZZcz 8245   ^cexp 9254

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 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 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 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212  df-iexp 9255

This theorem is referenced by:  expsubap  9302  expnegapd  9388