Theorem expnegap0 10568

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 9213	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnne0 8982	. . . . . . . . . 10 |- ( N e. NN -> N =/= 0 )
3	2	adantl 277	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> N =/= 0 )
4		nncn 8962	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
5	4	adantl 277	. . . . . . . . . . 11 |- ( ( A e. CC /\ N e. NN ) -> N e. CC )
6	5	negeq0d 8295	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> ( N = 0 <-> -u N = 0 ) )
7	6	necon3abid 2399	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N =/= 0 <-> -. -u N = 0 ) )
8	3, 7	mpbid 147	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. -u N = 0 )
9	8	iffalsed 3559	. . . . . . 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 9218	. . . . . . . . . . 11 |- ( N e. NN -> N e. NN0 )
11	10	adantl 277	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. NN0 )
12		nn0nlt0 9237	. . . . . . . . . 10 |- ( N e. NN0 -> -. N < 0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -. N < 0 )
14	11	nn0red 9265	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. RR )
15	14	lt0neg1d 8507	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N < 0 <-> 0 < -u N ) )
16	13, 15	mtbid 673	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. 0 < -u N )
17	16	iffalsed 3559	. . . . . . 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 8294	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -u -u N = N )
19	18	fveq2d 5541	. . . . . . . 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 5916	. . . . . . 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 2226	. . . . . 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 477	. . . . 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 999	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A e. CC )
24		simp3 1001	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. NN )
25	24	nnzd 9409	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. ZZ )
26	25	znegcld 9412	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> -u N e. ZZ )
27		simp2 1000	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A # 0 )
28	27	orcd 734	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( A # 0 \/ 0 <_ -u N ) )
29		exp3val 10562	. . . . . . 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 1249	. . . . . 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 1205	. . . . 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 10563	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
33	32	oveq2d 5916	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
34	33	adantlr 477	. . . . 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 2232	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
36		1div1e1 8696	. . . . . . 7 |- ( 1 / 1 ) = 1
37	36	eqcomi 2193	. . . . . 6 |- 1 = ( 1 / 1 )
38		negeq 8185	. . . . . . . . 9 |- ( N = 0 -> -u N = -u 0 )
39		neg0 8238	. . . . . . . . 9 |- -u 0 = 0
40	38, 39	eqtrdi 2238	. . . . . . . 8 |- ( N = 0 -> -u N = 0 )
41	40	oveq2d 5916	. . . . . . 7 |- ( N = 0 -> ( A ^ -u N ) = ( A ^ 0 ) )
42		exp0 10564	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
43	41, 42	sylan9eqr 2244	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = 1 )
44		oveq2 5908	. . . . . . . 8 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
45	44, 42	sylan9eqr 2244	. . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
46	45	oveq2d 5916	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( 1 / ( A ^ N ) ) = ( 1 / 1 ) )
47	37, 43, 46	3eqtr4a 2248	. . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
48	47	adantlr 477	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
49	35, 48	jaodan 798	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
50	1, 49	sylan2b 287	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
51	50	3impa 1196	1 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2160   =/= wne 2360   ifcif 3549   {csn 3610   class class class wbr 4021   X. cxp 4645   ` cfv 5238  (class class class)co 5900   CCcc 7844   0cc0 7846   1c1 7847   x. cmul 7851   < clt 8027   <_ cle 8028   -ucneg 8164   # cap 8573   / cdiv 8664   NNcn 8954   NN0cn0 9211   ZZcz 9288   seqcseq 10484   ^cexp 10559

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-n0 9212  df-z 9289  df-uz 9564  df-seqfrec 10485  df-exp 10560

This theorem is referenced by:  expineg2  10569  expn1ap0  10570  expnegzap  10594  efexp  11731  pcexp  12352  ex-exp  14965