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Theorem expneg 12156
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 10793 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 10564 . . . . . . . 8  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 464 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 10539 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 464 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 9914 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2700 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =/=  0  <->  -.  -u N  =  0
) )
83, 7mpbid 210 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  -u N  =  0 )
98iffalsed 3940 . . . . 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 10798 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  NN0 )
1110adantl 464 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
12 nn0nlt0 10818 . . . . . . . 8  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1311, 12syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1411nn0red 10849 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1514lt0neg1d 10118 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1613, 15mtbid 298 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
1716iffalsed 3940 . . . . 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 9913 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
1918fveq2d 5852 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2019oveq2d 6286 . . . . 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 2499 . . . 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 10863 . . . . 5  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
23 expval 12150 . . . . 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 472 . . . 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 12151 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2625oveq2d 6286 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
2721, 24, 263eqtr4d 2505 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
28 1div1e1 10233 . . . . 5  |-  ( 1  /  1 )  =  1
2928eqcomi 2467 . . . 4  |-  1  =  ( 1  / 
1 )
30 negeq 9803 . . . . . . 7  |-  ( N  =  0  ->  -u N  =  -u 0 )
31 neg0 9856 . . . . . . 7  |-  -u 0  =  0
3230, 31syl6eq 2511 . . . . . 6  |-  ( N  =  0  ->  -u N  =  0 )
3332oveq2d 6286 . . . . 5  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
34 exp0 12152 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3533, 34sylan9eqr 2517 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
36 oveq2 6278 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
3736, 34sylan9eqr 2517 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
3837oveq2d 6286 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
3929, 35, 383eqtr4a 2521 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4027, 39jaodan 783 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
411, 40sylan2b 473 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   ifcif 3929   {csn 4016   class class class wbr 4439    X. cxp 4986   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486    < clt 9617   -ucneg 9797    / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860    seqcseq 12089   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-seq 12090  df-exp 12149
This theorem is referenced by:  expneg2  12157  expn1  12158  expnegz  12182  efexp  13918  pcexp  14467  aaliou3lem8  22907  basellem3  23554  basellem4  23555  basellem8  23559  dvtan  30305  irrapxlem5  31001  pellexlem2  31005  nn0digval  33475
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