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

Theorem expnegap0 9428
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 8241 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 8018 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 266 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 7998 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 266 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 7377 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2259 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =/=  0  <->  -.  -u N  =  0
) )
83, 7mpbid 139 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  -u N  =  0 )
98iffalsed 3369 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  if ( -u N  =  0 ,  1 ,  if ( 0  <  -u N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) ) )  =  if ( 0  <  -u N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) ) )
10 nnnn0 8246 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  NN0 )
1110adantl 266 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
12 nn0nlt0 8265 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1311, 12syl 14 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1411nn0red 8293 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1514lt0neg1d 7581 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1613, 15mtbid 607 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
1716iffalsed 3369 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  if ( 0  <  -u N ,  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) )
185negnegd 7376 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
1918fveq2d 5210 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ,  CC ) `
 -u -u N )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ,  CC ) `
 N ) )
2019oveq2d 5556 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u -u N
) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N )
) )
219, 17, 203eqtrd 2092 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  if ( -u N  =  0 ,  1 ,  if ( 0  <  -u N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ) )
2221adantlr 454 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  if ( -u N  =  0 ,  1 ,  if ( 0  <  -u N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ) )
23 simp1 915 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A  e.  CC )
24 simp3 917 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  NN )
2524nnzd 8418 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  ZZ )
2625znegcld 8421 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  -u N  e.  ZZ )
27 simp2 916 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A #  0 )
2827orcd 662 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  ( A #  0  \/  0  <_ 
-u N ) )
29 expival 9422 . . . . . . 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 } ) ,  CC ) `  -u N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) ) ) )
3023, 26, 28, 29syl3anc 1146 . . . . . 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 } ) ,  CC ) `  -u N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) ) ) )
31303expa 1115 . . . . 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 } ) ,  CC ) `  -u N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u -u N
) ) ) ) )
32 expinnval 9423 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N )
)
3332oveq2d 5556 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ) )
3433adantlr 454 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ) )
3522, 31, 343eqtr4d 2098 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
36 1div1e1 7755 . . . . . . 7  |-  ( 1  /  1 )  =  1
3736eqcomi 2060 . . . . . 6  |-  1  =  ( 1  / 
1 )
38 negeq 7267 . . . . . . . . 9  |-  ( N  =  0  ->  -u N  =  -u 0 )
39 neg0 7320 . . . . . . . . 9  |-  -u 0  =  0
4038, 39syl6eq 2104 . . . . . . . 8  |-  ( N  =  0  ->  -u N  =  0 )
4140oveq2d 5556 . . . . . . 7  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
42 exp0 9424 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
4341, 42sylan9eqr 2110 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
44 oveq2 5548 . . . . . . . 8  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
4544, 42sylan9eqr 2110 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4645oveq2d 5556 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4737, 43, 463eqtr4a 2114 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4847adantlr 454 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4935, 48jaodan 721 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
501, 49sylan2b 275 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN0 )  -> 
( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
51503impa 1110 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 101    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409    =/= wne 2220   ifcif 3359   {csn 3403   class class class wbr 3792    X. cxp 4371   ` cfv 4930  (class class class)co 5540   CCcc 6945   0cc0 6947   1c1 6948    x. cmul 6952    < clt 7119    <_ cle 7120   -ucneg 7246   # cap 7646    / cdiv 7725   NNcn 7990   NN0cn0 8239   ZZcz 8302    seqcseq 9375   ^cexp 9419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-iseq 9376  df-iexp 9420
This theorem is referenced by:  expineg2  9429  expn1ap0  9430  expnegzap  9454
  Copyright terms: Public domain W3C validator