Theorem expnegap0 9484

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 8290	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnne0 8067	. . . . . . . . . 10 |- ( N e. NN -> N =/= 0 )
3	2	adantl 271	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> N =/= 0 )
4		nncn 8047	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
5	4	adantl 271	. . . . . . . . . . 11 |- ( ( A e. CC /\ N e. NN ) -> N e. CC )
6	5	negeq0d 7411	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> ( N = 0 <-> -u N = 0 ) )
7	6	necon3abid 2284	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N =/= 0 <-> -. -u N = 0 ) )
8	3, 7	mpbid 145	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. -u N = 0 )
9	8	iffalsed 3361	. . . . . . 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 8295	. . . . . . . . . . 11 |- ( N e. NN -> N e. NN0 )
11	10	adantl 271	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. NN0 )
12		nn0nlt0 8314	. . . . . . . . . 10 |- ( N e. NN0 -> -. N < 0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -. N < 0 )
14	11	nn0red 8342	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. RR )
15	14	lt0neg1d 7616	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> ( N < 0 <-> 0 < -u N ) )
16	13, 15	mtbid 629	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> -. 0 < -u N )
17	16	iffalsed 3361	. . . . . . 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 7410	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -u -u N = N )
19	18	fveq2d 5202	. . . . . . . 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 5548	. . . . . . 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 2117	. . . . . 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 460	. . . . 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 938	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A e. CC )
24		simp3 940	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. NN )
25	24	nnzd 8468	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. ZZ )
26	25	znegcld 8471	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> -u N e. ZZ )
27		simp2 939	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> A # 0 )
28	27	orcd 684	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( A # 0 \/ 0 <_ -u N ) )
29		expival 9478	. . . . . . 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 1169	. . . . . 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 1138	. . . . 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 9479	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) )
33	32	oveq2d 5548	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ) )
34	33	adantlr 460	. . . . 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 2123	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
36		1div1e1 7792	. . . . . . 7 |- ( 1 / 1 ) = 1
37	36	eqcomi 2085	. . . . . 6 |- 1 = ( 1 / 1 )
38		negeq 7301	. . . . . . . . 9 |- ( N = 0 -> -u N = -u 0 )
39		neg0 7354	. . . . . . . . 9 |- -u 0 = 0
40	38, 39	syl6eq 2129	. . . . . . . 8 |- ( N = 0 -> -u N = 0 )
41	40	oveq2d 5548	. . . . . . 7 |- ( N = 0 -> ( A ^ -u N ) = ( A ^ 0 ) )
42		exp0 9480	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
43	41, 42	sylan9eqr 2135	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = 1 )
44		oveq2 5540	. . . . . . . 8 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
45	44, 42	sylan9eqr 2135	. . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
46	45	oveq2d 5548	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( 1 / ( A ^ N ) ) = ( 1 / 1 ) )
47	37, 43, 46	3eqtr4a 2139	. . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
48	47	adantlr 460	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
49	35, 48	jaodan 743	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
50	1, 49	sylan2b 281	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
51	50	3impa 1133	1 |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   =/= wne 2245   ifcif 3351   {csn 3398   class class class wbr 3785   X. cxp 4361   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   x. cmul 6986   < clt 7153   <_ cle 7154   -ucneg 7280   # cap 7681   / cdiv 7760   NNcn 8039   NN0cn0 8288   ZZcz 8351   seqcseq 9431   ^cexp 9475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476

This theorem is referenced by:  expineg2  9485  expn1ap0  9486  expnegzap  9510