Theorem expnegap0 10690

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 9296	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnne0 9063	. . . . . . . . . 10 |- ( N e. NN -> N =/= 0 )
3	2	adantl 277	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> N =/= 0 )
4		nncn 9043	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
5	4	adantl 277	. . . . . . . . . . 11 |- ( ( A e. CC /\ N e. NN ) -> N e. CC )
6	5	negeq0d 8374	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> ( N = 0 <-> -u N = 0 ) )
7	6	necon3abid 2414	. . . . . . . . 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 3580	. . . . . . 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 9301	. . . . . . . . . . 11 |- ( N e. NN -> N e. NN0 )
11	10	adantl 277	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. NN0 )
12		nn0nlt0 9320	. . . . . . . . . 10 |- ( N e. NN0 -> -. N < 0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -. N < 0 )
14	11	nn0red 9348	. . . . . . . . . 10 |- ( ( A e. CC /\ N e. NN ) -> N e. RR )
15	14	lt0neg1d 8587	. . . . . . . . 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 3580	. . . . . . 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 8373	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> -u -u N = N )
19	18	fveq2d 5579	. . . . . . . 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 5959	. . . . . . 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 2241	. . . . . 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 9493	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> N e. ZZ )
26	25	znegcld 9496	. . . . . . 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 10684	. . . . . . 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 10685	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
33	32	oveq2d 5959	. . . . . 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 2247	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
36		1div1e1 8776	. . . . . . 7 |- ( 1 / 1 ) = 1
37	36	eqcomi 2208	. . . . . 6 |- 1 = ( 1 / 1 )
38		negeq 8264	. . . . . . . . 9 |- ( N = 0 -> -u N = -u 0 )
39		neg0 8317	. . . . . . . . 9 |- -u 0 = 0
40	38, 39	eqtrdi 2253	. . . . . . . 8 |- ( N = 0 -> -u N = 0 )
41	40	oveq2d 5959	. . . . . . 7 |- ( N = 0 -> ( A ^ -u N ) = ( A ^ 0 ) )
42		exp0 10686	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
43	41, 42	sylan9eqr 2259	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = 1 )
44		oveq2 5951	. . . . . . . 8 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
45	44, 42	sylan9eqr 2259	. . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
46	45	oveq2d 5959	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( 1 / ( A ^ N ) ) = ( 1 / 1 ) )
47	37, 43, 46	3eqtr4a 2263	. . . . 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 1372   e. wcel 2175   =/= wne 2375   ifcif 3570   {csn 3632   class class class wbr 4043   X. cxp 4672   ` cfv 5270  (class class class)co 5943   CCcc 7922   0cc0 7924   1c1 7925   x. cmul 7929   < clt 8106   <_ cle 8107   -ucneg 8243   # cap 8653   / cdiv 8744   NNcn 9035   NN0cn0 9294   ZZcz 9371   seqcseq 10590   ^cexp 10681

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-seqfrec 10591  df-exp 10682

This theorem is referenced by:  expineg2  10691  expn1ap0  10692  expnegzap  10716  efexp  11964  pcexp  12603  ex-exp  15625