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Theorem expneg 12143
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 10798 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 10569 . . . . . . . 8  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 466 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 10545 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 9923 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2713 . . . . . . 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 )
9 iffalse 3948 . . . . . 6  |-  ( -.  -u N  =  0  ->  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 ) ) ) )
108, 9syl 16 . . . . 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 ) ) ) )
11 nnnn0 10803 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  NN0 )
1211adantl 466 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
13 nn0nlt0 10823 . . . . . . . 8  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1412, 13syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1512nn0red 10854 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1615lt0neg1d 10123 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1714, 16mtbid 300 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
18 iffalse 3948 . . . . . 6  |-  ( -.  0  <  -u N  ->  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 ) ) )
1917, 18syl 16 . . . . 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 ) ) )
205negnegd 9922 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
2120fveq2d 5870 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2221oveq2d 6301 . . . . 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 ) ) )
2310, 19, 223eqtrd 2512 . . . 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 ) ) )
24 nnnegz 10868 . . . . 5  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
25 expval 12137 . . . . 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 ) ) ) ) )
2624, 25sylan2 474 . . . 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 ) ) ) ) )
27 expnnval 12138 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2827oveq2d 6301 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
2923, 26, 283eqtr4d 2518 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
30 1div1e1 10238 . . . . 5  |-  ( 1  /  1 )  =  1
3130eqcomi 2480 . . . 4  |-  1  =  ( 1  / 
1 )
32 negeq 9813 . . . . . . 7  |-  ( N  =  0  ->  -u N  =  -u 0 )
33 neg0 9866 . . . . . . 7  |-  -u 0  =  0
3432, 33syl6eq 2524 . . . . . 6  |-  ( N  =  0  ->  -u N  =  0 )
3534oveq2d 6301 . . . . 5  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
36 exp0 12139 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3735, 36sylan9eqr 2530 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
38 oveq2 6293 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
3938, 36sylan9eqr 2530 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4039oveq2d 6301 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4131, 37, 403eqtr4a 2534 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4229, 41jaodan 783 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
431, 42sylan2b 475 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 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939   {csn 4027   class class class wbr 4447    X. cxp 4997   ` cfv 5588  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    x. cmul 9498    < clt 9629   -ucneg 9807    / cdiv 10207   NNcn 10537   NN0cn0 10796   ZZcz 10865    seqcseq 12076   ^cexp 12135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-n0 10797  df-z 10866  df-seq 12077  df-exp 12136
This theorem is referenced by:  expneg2  12144  expn1  12145  expnegz  12169  efexp  13700  pcexp  14245  aaliou3lem8  22567  basellem3  23181  basellem4  23182  basellem8  23186  dvtan  29918  irrapxlem5  30593  pellexlem2  30597
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