Theorem expnegap0 9263

Description: Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)

Assertion

Ref	Expression
expnegap0	|- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )

Proof of Theorem expnegap0

Step	Hyp	Ref	Expression
1		elnn0 8183	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnne0 7942	. . . . . . . . . 10 |- ( N e. NN -> N =/= 0 )
3	2	adantl 262	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> N =/= 0 )
4		nncn 7922	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
5	4	adantl 262	. . . . . . . . . . 11 |- ( ( A e. CC /\ N e. NN ) -> N e. CC )
6	5	negeq0d 7314	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> ( N = 0 <-> -u N = 0 ) )
7	6	necon3abid 2244	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N =/= 0 <-> -. -u N = 0 ) )
8	3, 7	mpbid 135	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. -u N = 0 )
9	8	iffalsed 3341	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) ) = if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) )
10		nnnn0 8188	. . . . . . . . . . 11 |- ( N e. NN -> N e. NN0 )
11	10	adantl 262	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. NN0 )
12		nn0nlt0 8208	. . . . . . . . . 10 |- ( N e. NN0 -> -. N < 0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -. N < 0 )
14	11	nn0red 8236	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. RR )
15	14	lt0neg1d 7507	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N < 0 <-> 0 < -u N ) )
16	13, 15	mtbid 597	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. 0 < -u N )
17	16	iffalsed 3341	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) )
18	5	negnegd 7313	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -u -u N = N )
19	18	fveq2d 5182	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) )
20	19	oveq2d 5528	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ) )
21	9, 17, 20	3eqtrd 2076	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ) )
22	21	adantlr 446	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ) )
23		simp1 904	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A e. CC )
24		simp3 906	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. NN )
25	24	nnzd 8359	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. ZZ )
26	25	znegcld 8362	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> -u N e. ZZ )
27		simp2 905	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A # 0 )
28	27	orcd 652	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( A # 0 \/ 0 <_ -u N ) )
29		expival 9257	. . . . . . 7 |- ( ( A e. CC /\ -u N e. ZZ /\ ( A # 0 \/ 0 <_ -u N ) ) -> ( A ^ -u N ) = if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) ) )
30	23, 26, 28, 29	syl3anc 1135	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( A ^ -u N ) = if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) ) )
31	30	3expa 1104	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( A ^ -u N ) = if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u -u N ) ) ) ) )
32		expinnval 9258	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) )
33	32	oveq2d 5528	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ) )
34	33	adantlr 446	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ) )
35	22, 31, 34	3eqtr4d 2082	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
36		1div1e1 7681	. . . . . . 7 |- ( 1 / 1 ) = 1
37	36	eqcomi 2044	. . . . . 6 |- 1 = ( 1 / 1 )
38		negeq 7204	. . . . . . . . 9 |- ( N = 0 -> -u N = -u 0 )
39		neg0 7257	. . . . . . . . 9 |- -u 0 = 0
40	38, 39	syl6eq 2088	. . . . . . . 8 |- ( N = 0 -> -u N = 0 )
41	40	oveq2d 5528	. . . . . . 7 |- ( N = 0 -> ( A ^ -u N ) = ( A ^ 0 ) )
42		exp0 9259	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
43	41, 42	sylan9eqr 2094	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = 1 )
44		oveq2 5520	. . . . . . . 8 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
45	44, 42	sylan9eqr 2094	. . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
46	45	oveq2d 5528	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( 1 / ( A ^ N ) ) = ( 1 / 1 ) )
47	37, 43, 46	3eqtr4a 2098	. . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
48	47	adantlr 446	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
49	35, 48	jaodan 710	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
50	1, 49	sylan2b 271	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
51	50	3impa 1099	1 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   =/= wne 2204   ifcif 3331   {csn 3375   class class class wbr 3764   X. cxp 4343   ` cfv 4902  (class class class)co 5512   CCcc 6887   0cc0 6889   1c1 6890   x. cmul 6894   < clt 7060   <_ cle 7061   -ucneg 7183   # cap 7572   / cdiv 7651   NNcn 7914   NN0cn0 8181   ZZcz 8245   seqcseq 9211   ^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:  expineg2  9264  expn1ap0  9265  expnegzap  9289