ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expnegap0 Unicode version

Theorem expnegap0 10313
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 8991 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 8760 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 275 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 8740 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 275 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 8077 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2347 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =/=  0  <->  -.  -u N  =  0
) )
83, 7mpbid 146 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  -u N  =  0 )
98iffalsed 3484 . . . . . . 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 8996 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  NN0 )
1110adantl 275 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
12 nn0nlt0 9015 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1311, 12syl 14 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1411nn0red 9043 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1514lt0neg1d 8289 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1613, 15mtbid 661 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
1716iffalsed 3484 . . . . . . 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 8076 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
1918fveq2d 5425 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2019oveq2d 5790 . . . . . . 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 2176 . . . . . 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 468 . . . . 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 981 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A  e.  CC )
24 simp3 983 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  NN )
2524nnzd 9184 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  ZZ )
2625znegcld 9187 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  -u N  e.  ZZ )
27 simp2 982 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A #  0 )
2827orcd 722 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  ( A #  0  \/  0  <_ 
-u N ) )
29 exp3val 10307 . . . . . . 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 1216 . . . . . 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 1181 . . . . 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 10308 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
3332oveq2d 5790 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
3433adantlr 468 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
3522, 31, 343eqtr4d 2182 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
36 1div1e1 8476 . . . . . . 7  |-  ( 1  /  1 )  =  1
3736eqcomi 2143 . . . . . 6  |-  1  =  ( 1  / 
1 )
38 negeq 7967 . . . . . . . . 9  |-  ( N  =  0  ->  -u N  =  -u 0 )
39 neg0 8020 . . . . . . . . 9  |-  -u 0  =  0
4038, 39syl6eq 2188 . . . . . . . 8  |-  ( N  =  0  ->  -u N  =  0 )
4140oveq2d 5790 . . . . . . 7  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
42 exp0 10309 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
4341, 42sylan9eqr 2194 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
44 oveq2 5782 . . . . . . . 8  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
4544, 42sylan9eqr 2194 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4645oveq2d 5790 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4737, 43, 463eqtr4a 2198 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4847adantlr 468 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4935, 48jaodan 786 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
501, 49sylan2b 285 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN0 )  -> 
( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
51503impa 1176 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 103    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480    =/= wne 2308   ifcif 3474   {csn 3527   class class class wbr 3929    X. cxp 4537   ` cfv 5123  (class class class)co 5774   CCcc 7630   0cc0 7632   1c1 7633    x. cmul 7637    < clt 7812    <_ cle 7813   -ucneg 7946   # cap 8355    / cdiv 8444   NNcn 8732   NN0cn0 8989   ZZcz 9066    seqcseq 10230   ^cexp 10304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-seqfrec 10231  df-exp 10305
This theorem is referenced by:  expineg2  10314  expn1ap0  10315  expnegzap  10339  efexp  11400  ex-exp  12998
  Copyright terms: Public domain W3C validator