Theorem expnegap0 10459

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 9112	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnne0 8881	. . . . . . . . . 10 |- ( N e. NN -> N =/= 0 )
3	2	adantl 275	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> N =/= 0 )
4		nncn 8861	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
5	4	adantl 275	. . . . . . . . . . 11 |- ( ( A e. CC /\ N e. NN ) -> N e. CC )
6	5	negeq0d 8197	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> ( N = 0 <-> -u N = 0 ) )
7	6	necon3abid 2374	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N =/= 0 <-> -. -u N = 0 ) )
8	3, 7	mpbid 146	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. -u N = 0 )
9	8	iffalsed 3529	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) ) = if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) )
10		nnnn0 9117	. . . . . . . . . . 11 |- ( N e. NN -> N e. NN0 )
11	10	adantl 275	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. NN0 )
12		nn0nlt0 9136	. . . . . . . . . 10 |- ( N e. NN0 -> -. N < 0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -. N < 0 )
14	11	nn0red 9164	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. RR )
15	14	lt0neg1d 8409	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N < 0 <-> 0 < -u N ) )
16	13, 15	mtbid 662	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. 0 < -u N )
17	16	iffalsed 3529	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) )
18	5	negnegd 8196	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -u -u N = N )
19	18	fveq2d 5489	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
20	19	oveq2d 5857	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
21	9, 17, 20	3eqtrd 2202	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
22	21	adantlr 469	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> if ( -u N = 0 , 1 , if ( 0 < -u N , ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
23		simp1 987	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A e. CC )
24		simp3 989	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. NN )
25	24	nnzd 9308	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. ZZ )
26	25	znegcld 9311	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> -u N e. ZZ )
27		simp2 988	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A # 0 )
28	27	orcd 723	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( A # 0 \/ 0 <_ -u N ) )
29		exp3val 10453	. . . . . . 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 } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) ) )
30	23, 26, 28, 29	syl3anc 1228	. . . . . 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 } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) ) )
31	30	3expa 1193	. . . . 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 } ) ) ` -u N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) ) ) )
32		expnnval 10454	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
33	32	oveq2d 5857	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
34	33	adantlr 469	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
35	22, 31, 34	3eqtr4d 2208	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
36		1div1e1 8596	. . . . . . 7 |- ( 1 / 1 ) = 1
37	36	eqcomi 2169	. . . . . 6 |- 1 = ( 1 / 1 )
38		negeq 8087	. . . . . . . . 9 |- ( N = 0 -> -u N = -u 0 )
39		neg0 8140	. . . . . . . . 9 |- -u 0 = 0
40	38, 39	eqtrdi 2214	. . . . . . . 8 |- ( N = 0 -> -u N = 0 )
41	40	oveq2d 5857	. . . . . . 7 |- ( N = 0 -> ( A ^ -u N ) = ( A ^ 0 ) )
42		exp0 10455	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
43	41, 42	sylan9eqr 2220	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = 1 )
44		oveq2 5849	. . . . . . . 8 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
45	44, 42	sylan9eqr 2220	. . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
46	45	oveq2d 5857	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( 1 / ( A ^ N ) ) = ( 1 / 1 ) )
47	37, 43, 46	3eqtr4a 2224	. . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
48	47	adantlr 469	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
49	35, 48	jaodan 787	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
50	1, 49	sylan2b 285	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
51	50	3impa 1184	1 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   ifcif 3519   {csn 3575   class class class wbr 3981   X. cxp 4601   ` cfv 5187  (class class class)co 5841   CCcc 7747   0cc0 7749   1c1 7750   x. cmul 7754   < clt 7929   <_ cle 7930   -ucneg 8066   # cap 8475   / cdiv 8564   NNcn 8853   NN0cn0 9110   ZZcz 9187   seqcseq 10376   ^cexp 10450

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-n0 9111  df-z 9188  df-uz 9463  df-seqfrec 10377  df-exp 10451

This theorem is referenced by:  expineg2  10460  expn1ap0  10461  expnegzap  10485  efexp  11619  pcexp  12237  ex-exp  13568