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Theorem expnegap0 10568
Description: Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
Assertion
Ref Expression
expnegap0  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e. 
NN0 )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )

Proof of Theorem expnegap0
StepHypRef Expression
1 elnn0 9213 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 8982 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 277 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 8962 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 277 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 8295 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2399 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =/=  0  <->  -.  -u N  =  0
) )
83, 7mpbid 147 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  -u N  =  0 )
98iffalsed 3559 . . . . . . 7  |-  ( ( 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 9218 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  NN0 )
1110adantl 277 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
12 nn0nlt0 9237 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1311, 12syl 14 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1411nn0red 9265 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1514lt0neg1d 8507 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1613, 15mtbid 673 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
1716iffalsed 3559 . . . . . . 7  |-  ( ( 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 8294 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
1918fveq2d 5541 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2019oveq2d 5916 . . . . . . 7  |-  ( ( 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 2226 . . . . . 6  |-  ( ( 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 ) ) )
2221adantlr 477 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  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 ) ) )
23 simp1 999 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A  e.  CC )
24 simp3 1001 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  NN )
2524nnzd 9409 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  ZZ )
2625znegcld 9412 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  -u N  e.  ZZ )
27 simp2 1000 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A #  0 )
2827orcd 734 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  ( A #  0  \/  0  <_ 
-u N ) )
29 exp3val 10562 . . . . . . 7  |-  ( ( A  e.  CC  /\  -u N  e.  ZZ  /\  ( A #  0  \/  0  <_  -u N ) )  ->  ( 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 ) ) ) ) )
3023, 26, 28, 29syl3anc 1249 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  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 ) ) ) ) )
31303expa 1205 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  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 ) ) ) ) )
32 expnnval 10563 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
3332oveq2d 5916 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
3433adantlr 477 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
3522, 31, 343eqtr4d 2232 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
36 1div1e1 8696 . . . . . . 7  |-  ( 1  /  1 )  =  1
3736eqcomi 2193 . . . . . 6  |-  1  =  ( 1  / 
1 )
38 negeq 8185 . . . . . . . . 9  |-  ( N  =  0  ->  -u N  =  -u 0 )
39 neg0 8238 . . . . . . . . 9  |-  -u 0  =  0
4038, 39eqtrdi 2238 . . . . . . . 8  |-  ( N  =  0  ->  -u N  =  0 )
4140oveq2d 5916 . . . . . . 7  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
42 exp0 10564 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
4341, 42sylan9eqr 2244 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
44 oveq2 5908 . . . . . . . 8  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
4544, 42sylan9eqr 2244 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4645oveq2d 5916 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4737, 43, 463eqtr4a 2248 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4847adantlr 477 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4935, 48jaodan 798 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
501, 49sylan2b 287 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN0 )  -> 
( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
51503impa 1196 1  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e. 
NN0 )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709    /\ w3a 980    = wceq 1364    e. wcel 2160    =/= wne 2360   ifcif 3549   {csn 3610   class class class wbr 4021    X. cxp 4645   ` cfv 5238  (class class class)co 5900   CCcc 7844   0cc0 7846   1c1 7847    x. cmul 7851    < clt 8027    <_ cle 8028   -ucneg 8164   # cap 8573    / cdiv 8664   NNcn 8954   NN0cn0 9211   ZZcz 9288    seqcseq 10484   ^cexp 10559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-n0 9212  df-z 9289  df-uz 9564  df-seqfrec 10485  df-exp 10560
This theorem is referenced by:  expineg2  10569  expn1ap0  10570  expnegzap  10594  efexp  11731  pcexp  12352  ex-exp  14965
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