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Theorem expneg 11873
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 10581 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 10354 . . . . . . . 8  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 466 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 10330 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 9711 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2641 . . . . . . 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 3799 . . . . . 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 10586 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  NN0 )
1211adantl 466 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
13 nn0nlt0 10606 . . . . . . . 8  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1412, 13syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1512nn0red 10637 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1615lt0neg1d 9909 . . . . . . 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 3799 . . . . . 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 9710 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
2120fveq2d 5695 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2221oveq2d 6107 . . . . 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 2479 . . . 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 10649 . . . . 5  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
25 expval 11867 . . . . 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 11868 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2827oveq2d 6107 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
2923, 26, 283eqtr4d 2485 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
30 1div1e1 10024 . . . . 5  |-  ( 1  /  1 )  =  1
3130eqcomi 2447 . . . 4  |-  1  =  ( 1  / 
1 )
32 negeq 9602 . . . . . . 7  |-  ( N  =  0  ->  -u N  =  -u 0 )
33 neg0 9655 . . . . . . 7  |-  -u 0  =  0
3432, 33syl6eq 2491 . . . . . 6  |-  ( N  =  0  ->  -u N  =  0 )
3534oveq2d 6107 . . . . 5  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
36 exp0 11869 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3735, 36sylan9eqr 2497 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
38 oveq2 6099 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
3938, 36sylan9eqr 2497 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4039oveq2d 6107 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4131, 37, 403eqtr4a 2501 . . 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 1369    e. wcel 1756    =/= wne 2606   ifcif 3791   {csn 3877   class class class wbr 4292    X. cxp 4838   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283    x. cmul 9287    < clt 9418   -ucneg 9596    / cdiv 9993   NNcn 10322   NN0cn0 10579   ZZcz 10646    seqcseq 11806   ^cexp 11865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-seq 11807  df-exp 11866
This theorem is referenced by:  expneg2  11874  expn1  11875  expnegz  11898  efexp  13385  pcexp  13926  aaliou3lem8  21811  basellem3  22420  basellem4  22421  basellem8  22425  dvtan  28442  irrapxlem5  29167  pellexlem2  29171
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