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Theorem expnegap0 10692
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 9297 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnne0 9064 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
32adantl 277 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  =/=  0 )
4 nncn 9044 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
54adantl 277 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  CC )
65negeq0d 8375 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  =  0  <->  -u N  =  0
) )
76necon3abid 2415 . . . . . . . . 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 3581 . . . . . . 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 9302 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  NN0 )
1110adantl 277 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN0 )
12 nn0nlt0 9321 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  -.  N  <  0 )
1311, 12syl 14 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  N  <  0
)
1411nn0red 9349 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  RR )
1514lt0neg1d 8588 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( N  <  0  <->  0  <  -u N ) )
1613, 15mtbid 674 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  -.  0  <  -u N
)
1716iffalsed 3581 . . . . . . 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 8374 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN )  -> 
-u -u N  =  N )
1918fveq2d 5580 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u -u N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2019oveq2d 5960 . . . . . . 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 2242 . . . . . 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 1000 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A  e.  CC )
24 simp3 1002 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  NN )
2524nnzd 9494 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  N  e.  ZZ )
2625znegcld 9497 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  -u N  e.  ZZ )
27 simp2 1001 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  A #  0 )
2827orcd 735 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  ( A #  0  \/  0  <_ 
-u N ) )
29 exp3val 10686 . . . . . . 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 1250 . . . . . 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 1206 . . . . 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 10687 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
3332oveq2d 5960 . . . . . 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 2248 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
36 1div1e1 8777 . . . . . . 7  |-  ( 1  /  1 )  =  1
3736eqcomi 2209 . . . . . 6  |-  1  =  ( 1  / 
1 )
38 negeq 8265 . . . . . . . . 9  |-  ( N  =  0  ->  -u N  =  -u 0 )
39 neg0 8318 . . . . . . . . 9  |-  -u 0  =  0
4038, 39eqtrdi 2254 . . . . . . . 8  |-  ( N  =  0  ->  -u N  =  0 )
4140oveq2d 5960 . . . . . . 7  |-  ( N  =  0  ->  ( A ^ -u N )  =  ( A ^
0 ) )
42 exp0 10688 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
4341, 42sylan9eqr 2260 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ -u N )  =  1 )
44 oveq2 5952 . . . . . . . 8  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
4544, 42sylan9eqr 2260 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
4645oveq2d 5960 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( 1  / 
( A ^ N
) )  =  ( 1  /  1 ) )
4737, 43, 463eqtr4a 2264 . . . . 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 799 . . 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 1197 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 710    /\ w3a 981    = wceq 1373    e. wcel 2176    =/= wne 2376   ifcif 3571   {csn 3633   class class class wbr 4044    X. cxp 4673   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   1c1 7926    x. cmul 7930    < clt 8107    <_ cle 8108   -ucneg 8244   # cap 8654    / cdiv 8745   NNcn 9036   NN0cn0 9295   ZZcz 9372    seqcseq 10592   ^cexp 10683
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-seqfrec 10593  df-exp 10684
This theorem is referenced by:  expineg2  10693  expn1ap0  10694  expnegzap  10718  efexp  11993  pcexp  12632  ex-exp  15663
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