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Theorem expneg 11127
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 9983 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 9794 . . . . . . . 8  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 452 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 9770 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 452 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 9165 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2492 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =/=  0  <->  -.  -u N  =  0
) )
83, 7mpbid 201 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  -u N  =  0 )
9 iffalse 3585 . . . . . 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 15 . . . . 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 9988 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  NN0 )
1211adantl 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
13 nn0nlt0 10008 . . . . . . . 8  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1412, 13syl 15 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1512nn0red 10035 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1615lt0neg1d 9358 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1714, 16mtbid 291 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
18 iffalse 3585 . . . . . 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 15 . . . . 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 9164 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
2120fveq2d 5545 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2221oveq2d 5890 . . . . 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 2332 . . . 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 10043 . . . . 5  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
25 expval 11122 . . . . 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 460 . . . 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 11123 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2827oveq2d 5890 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ) )
2923, 26, 283eqtr4d 2338 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
30 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
3130div1i 9504 . . . . 5  |-  ( 1  /  1 )  =  1
3231eqcomi 2300 . . . 4  |-  1  =  ( 1  / 
1 )
33 negeq 9060 . . . . . . 7  |-  ( N  =  0  ->  -u N  =  -u 0 )
34 neg0 9109 . . . . . . 7  |-  -u 0  =  0
3533, 34syl6eq 2344 . . . . . 6  |-  ( N  =  0  ->  -u N  =  0 )
3635oveq2d 5890 . . . . 5  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
37 exp0 11124 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3836, 37sylan9eqr 2350 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
39 oveq2 5882 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
4039, 37sylan9eqr 2350 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4140oveq2d 5890 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4232, 38, 413eqtr4a 2354 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4329, 42jaodan 760 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
441, 43sylan2b 461 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 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578   {csn 3653   class class class wbr 4039    X. cxp 4703   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040    seq cseq 11062   ^cexp 11120
This theorem is referenced by:  expneg2  11128  expn1  11129  expnegz  11152  efexp  12397  pcexp  12928  aaliou3lem8  19741  basellem3  20336  basellem4  20337  basellem8  20341  irrapxlem5  27014  pellexlem2  27018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-seq 11063  df-exp 11121
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