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Theorem expnegap0 10078
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 8773 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 8548 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 272 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 8528 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 272 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 7882 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2301 . . . . . . . . 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 3423 . . . . . . 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 8778 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  NN0 )
1110adantl 272 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
12 nn0nlt0 8797 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1311, 12syl 14 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1411nn0red 8825 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1514lt0neg1d 8090 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1613, 15mtbid 635 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
1716iffalsed 3423 . . . . . . 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 7881 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
1918fveq2d 5344 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2019oveq2d 5706 . . . . . . 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 2131 . . . . . 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 462 . . . . 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 946 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A  e.  CC )
24 simp3 948 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  NN )
2524nnzd 8966 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  ZZ )
2625znegcld 8969 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  -u N  e.  ZZ )
27 simp2 947 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A #  0 )
2827orcd 690 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  ( A #  0  \/  0  <_ 
-u N ) )
29 exp3val 10072 . . . . . . 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 1181 . . . . . 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 1146 . . . . 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 10073 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
3332oveq2d 5706 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
3433adantlr 462 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( 1  /  ( A ^ N ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ) )
3522, 31, 343eqtr4d 2137 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
36 1div1e1 8268 . . . . . . 7  |-  ( 1  /  1 )  =  1
3736eqcomi 2099 . . . . . 6  |-  1  =  ( 1  / 
1 )
38 negeq 7772 . . . . . . . . 9  |-  ( N  =  0  ->  -u N  =  -u 0 )
39 neg0 7825 . . . . . . . . 9  |-  -u 0  =  0
4038, 39syl6eq 2143 . . . . . . . 8  |-  ( N  =  0  ->  -u N  =  0 )
4140oveq2d 5706 . . . . . . 7  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
42 exp0 10074 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
4341, 42sylan9eqr 2149 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
44 oveq2 5698 . . . . . . . 8  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
4544, 42sylan9eqr 2149 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4645oveq2d 5706 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4737, 43, 463eqtr4a 2153 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4847adantlr 462 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  =  0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
4935, 48jaodan 749 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
501, 49sylan2b 282 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN0 )  -> 
( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
51503impa 1141 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 667    /\ w3a 927    = wceq 1296    e. wcel 1445    =/= wne 2262   ifcif 3413   {csn 3466   class class class wbr 3867    X. cxp 4465   ` cfv 5049  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448    x. cmul 7452    < clt 7619    <_ cle 7620   -ucneg 7751   # cap 8155    / cdiv 8236   NNcn 8520   NN0cn0 8771   ZZcz 8848    seqcseq 10000   ^cexp 10069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-seqfrec 10001  df-exp 10070
This theorem is referenced by:  expineg2  10079  expn1ap0  10080  expnegzap  10104  efexp  11121  ex-exp  12362
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