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Theorem expneg 11381
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 10215 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 10024 . . . . . . . 8  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 453 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 10000 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 453 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 9395 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2631 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =/=  0  <->  -.  -u N  =  0
) )
83, 7mpbid 202 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  -u N  =  0 )
9 iffalse 3738 . . . . . 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 10220 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  NN0 )
1211adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
13 nn0nlt0 10240 . . . . . . . 8  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1412, 13syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1512nn0red 10267 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1615lt0neg1d 9588 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1714, 16mtbid 292 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
18 iffalse 3738 . . . . . 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 9394 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
2120fveq2d 5724 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2221oveq2d 6089 . . . . 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 2471 . . . 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 10277 . . . . 5  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
25 expval 11376 . . . . 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 461 . . . 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 11377 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2827oveq2d 6089 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ) )
2923, 26, 283eqtr4d 2477 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
30 ax-1cn 9040 . . . . . 6  |-  1  e.  CC
3130div1i 9734 . . . . 5  |-  ( 1  /  1 )  =  1
3231eqcomi 2439 . . . 4  |-  1  =  ( 1  / 
1 )
33 negeq 9290 . . . . . . 7  |-  ( N  =  0  ->  -u N  =  -u 0 )
34 neg0 9339 . . . . . . 7  |-  -u 0  =  0
3533, 34syl6eq 2483 . . . . . 6  |-  ( N  =  0  ->  -u N  =  0 )
3635oveq2d 6089 . . . . 5  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
37 exp0 11378 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3836, 37sylan9eqr 2489 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
39 oveq2 6081 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
4039, 37sylan9eqr 2489 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4140oveq2d 6089 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4232, 38, 413eqtr4a 2493 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4329, 42jaodan 761 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
441, 43sylan2b 462 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   ifcif 3731   {csn 3806   class class class wbr 4204    X. cxp 4868   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112   -ucneg 9284    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274    seq cseq 11315   ^cexp 11374
This theorem is referenced by:  expneg2  11382  expn1  11383  expnegz  11406  efexp  12694  pcexp  13225  aaliou3lem8  20254  basellem3  20857  basellem4  20858  basellem8  20862  irrapxlem5  26880  pellexlem2  26884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-seq 11316  df-exp 11375
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