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Theorem expneg 12868
Description: Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expneg |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
Proof of Theorem expneg
StepHypRef Expression
1 elnn0 11294 . 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2 nnne0 11053 . . . . . . . 8 |- ( N e. NN -> N =/= 0 )
32adantl 482 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> N =/= 0 )
4 nncn 11028 . . . . . . . . . 10 |- ( N e. NN -> N e. CC )
54adantl 482 . . . . . . . . 9 |- ( ( A e. CC /\ N e. NN ) -> N e. CC )
65negeq0d 10384 . . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> ( N = 0 <-> -u N = 0 ) )
76necon3abid 2830 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( N =/= 0 <-> -. -u N = 0 ) )
83, 7mpbid 222 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> -. -u N = 0 )
98iffalsed 4097 . . . . 5 |- ( ( 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 11299 . . . . . . . . 9 |- ( N e. NN -> N e. NN0 )
1110adantl 482 . . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> N e. NN0 )
12 nn0nlt0 11319 . . . . . . . 8 |- ( N e. NN0 -> -. N < 0 )
1311, 12syl 17 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> -. N < 0 )
1411nn0red 11352 . . . . . . . 8 |- ( ( A e. CC /\ N e. NN ) -> N e. RR )
1514lt0neg1d 10597 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> ( N < 0 <-> 0 < -u N ) )
1613, 15mtbid 314 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> -. 0 < -u N )
1716iffalsed 4097 . . . . 5 |- ( ( 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 ) ) )
185negnegd 10383 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> -u -u N = N )
1918fveq2d 6195 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
2019oveq2d 6666 . . . . 5 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u -u N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
219, 17, 203eqtrd 2660 . . . 4 |- ( ( 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 nnnegz 11380 . . . . 5 |- ( N e. NN -> -u N e. ZZ )
23 expval 12862 . . . . 5 |- ( ( A e. CC /\ -u N e. ZZ ) -> ( 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 ) ) ) ) )
2422, 23sylan2 491 . . . 4 |- ( ( A e. CC /\ 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 ) ) ) ) )
25 expnnval 12863 . . . . 5 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
2625oveq2d 6666 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( 1 / ( A ^ N ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) ) )
2721, 24, 263eqtr4d 2666 . . 3 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
28 1div1e1 10717 . . . . 5 |- ( 1 / 1 ) = 1
2928eqcomi 2631 . . . 4 |- 1 = ( 1 / 1 )
30 negeq 10273 . . . . . . 7 |- ( N = 0 -> -u N = -u 0 )
31 neg0 10327 . . . . . . 7 |- -u 0 = 0
3230, 31syl6eq 2672 . . . . . 6 |- ( N = 0 -> -u N = 0 )
3332oveq2d 6666 . . . . 5 |- ( N = 0 -> ( A ^ -u N ) = ( A ^ 0 ) )
34 exp0 12864 . . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
3533, 34sylan9eqr 2678 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = 1 )
36 oveq2 6658 . . . . . 6 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
3736, 34sylan9eqr 2678 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
3837oveq2d 6666 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( 1 / ( A ^ N ) ) = ( 1 / 1 ) )
3929, 35, 383eqtr4a 2682 . . 3 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
4027, 39jaodan 826 . 2 |- ( ( A e. CC /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
411, 40sylan2b 492 1 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   {csn 4177   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   seqcseq 12801   ^cexp 12860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-seq 12802  df-exp 12861
This theorem is referenced by:  expneg2  12869  expn1  12870  expnegz  12894  efexp  14831  pcexp  15564  aaliou3lem8  24100  basellem3  24809  basellem4  24810  basellem8  24814  ex-exp  27307  dvtan  33460  irrapxlem5  37390  pellexlem2  37394  nn0digval  42394
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