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Theorem expnegzap 10357
Description: Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expnegzap  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )

Proof of Theorem expnegzap
StepHypRef Expression
1 elznn0 9092 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2 expnegap0 10331 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e. 
NN0 )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
323expia 1184 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0 )  ->  ( N  e.  NN0  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) ) )
43adantr 274 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  RR )  ->  ( N  e.  NN0  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) ) )
5 simpl 108 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  ( A  e.  CC  /\  A #  0 ) )
6 simprl 521 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  N  e.  RR )
76recnd 7817 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  N  e.  CC )
8 simprr 522 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  -u N  e.  NN0 )
9 expineg2 10332 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
105, 7, 8, 9syl12anc 1215 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
1110oveq2d 5797 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  (
1  /  ( A ^ N ) )  =  ( 1  / 
( 1  /  ( A ^ -u N ) ) ) )
12 expcl 10341 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ -u N
)  e.  CC )
1312ad2ant2rl 503 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  ( A ^ -u N )  e.  CC )
14 simpll 519 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  A  e.  CC )
15 simplr 520 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  A #  0 )
168nn0zd 9194 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  -u N  e.  ZZ )
17 expap0i 10355 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  -u N  e.  ZZ )  ->  ( A ^ -u N ) #  0 )
1814, 15, 16, 17syl3anc 1217 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  ( A ^ -u N ) #  0 )
1913, 18recrecapd 8568 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  (
1  /  ( 1  /  ( A ^ -u N ) ) )  =  ( A ^ -u N ) )
2011, 19eqtr2d 2174 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN0 ) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
2120expr 373 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  RR )  ->  ( -u N  e. 
NN0  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) ) )
224, 21jaod 707 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  RR )  ->  ( ( N  e. 
NN0  \/  -u N  e. 
NN0 )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) ) )
2322expimpd 361 . . 3  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) ) )
241, 23syl5bi 151 . 2  |-  ( ( A  e.  CC  /\  A #  0 )  ->  ( N  e.  ZZ  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) ) )
25243impia 1179 1  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3936  (class class class)co 5781   CCcc 7641   RRcr 7642   0cc0 7643   1c1 7644   -ucneg 7957   # cap 8366    / cdiv 8455   NN0cn0 9000   ZZcz 9077   ^cexp 10322
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-seqfrec 10249  df-exp 10323
This theorem is referenced by:  expsubap  10371  expnegapd  10461
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