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Theorem oexpneg 12838
Description: The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)
Assertion
Ref Expression
oexpneg  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( -u A ^ N
)  =  -u ( A ^ N ) )

Proof of Theorem oexpneg
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 nnz 10235 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
2 odd2np1 12835 . . . . 5  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
31, 2syl 16 . . . 4  |-  ( N  e.  NN  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
43biimpa 471 . . 3  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
543adant1 975 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
6 simpl1 960 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  A  e.  CC )
7 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( 2  x.  n )  +  1 )  =  N )
8 simpl2 961 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  NN )
98nncnd 9948 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  CC )
10 ax-1cn 8981 . . . . . . . . . 10  |-  1  e.  CC
1110a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
1  e.  CC )
12 2z 10244 . . . . . . . . . . 11  |-  2  e.  ZZ
13 simprl 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  ZZ )
14 zmulcl 10256 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2  x.  n
)  e.  ZZ )
1512, 13, 14sylancr 645 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  ZZ )
1615zcnd 10308 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  CC )
179, 11, 16subadd2d 9362 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( N  - 
1 )  =  ( 2  x.  n )  <-> 
( ( 2  x.  n )  +  1 )  =  N ) )
187, 17mpbird 224 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( N  -  1 )  =  ( 2  x.  n ) )
19 nnm1nn0 10193 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
208, 19syl 16 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( N  -  1 )  e.  NN0 )
2118, 20eqeltrrd 2462 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  NN0 )
226, 21expcld 11450 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
2  x.  n ) )  e.  CC )
2322, 6mulneg2d 9419 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( A ^
( 2  x.  n
) )  x.  -u A
)  =  -u (
( A ^ (
2  x.  n ) )  x.  A ) )
24 sqneg 11369 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
256, 24syl 16 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ 2 )  =  ( A ^ 2 ) )
2625oveq1d 6035 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( -u A ^ 2 ) ^
n )  =  ( ( A ^ 2 ) ^ n ) )
276negcld 9330 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  -u A  e.  CC )
28 2re 10001 . . . . . . . . . . 11  |-  2  e.  RR
2928a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
2  e.  RR )
3013zred 10307 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  RR )
31 2pos 10014 . . . . . . . . . . 11  |-  0  <  2
3231a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <  2 )
3321nn0ge0d 10209 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  ( 2  x.  n ) )
34 prodge0 9789 . . . . . . . . . 10  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
3529, 30, 32, 33, 34syl22anc 1185 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  n )
36 elnn0z 10226 . . . . . . . . 9  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
3713, 35, 36sylanbrc 646 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  NN0 )
38 2nn0 10170 . . . . . . . . 9  |-  2  e.  NN0
3938a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
2  e.  NN0 )
4027, 37, 39expmuld 11453 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ (
2  x.  n ) )  =  ( (
-u A ^ 2 ) ^ n ) )
416, 37, 39expmuld 11453 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
2  x.  n ) )  =  ( ( A ^ 2 ) ^ n ) )
4226, 40, 413eqtr4d 2429 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ (
2  x.  n ) )  =  ( A ^ ( 2  x.  n ) ) )
4342oveq1d 6035 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( -u A ^ ( 2  x.  n ) )  x.  -u A )  =  ( ( A ^ (
2  x.  n ) )  x.  -u A
) )
4427, 21expp1d 11451 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ (
( 2  x.  n
)  +  1 ) )  =  ( (
-u A ^ (
2  x.  n ) )  x.  -u A
) )
457oveq2d 6036 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ (
( 2  x.  n
)  +  1 ) )  =  ( -u A ^ N ) )
4644, 45eqtr3d 2421 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( -u A ^ ( 2  x.  n ) )  x.  -u A )  =  (
-u A ^ N
) )
4743, 46eqtr3d 2421 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( A ^
( 2  x.  n
) )  x.  -u A
)  =  ( -u A ^ N ) )
4823, 47eqtr3d 2421 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  -u ( ( A ^
( 2  x.  n
) )  x.  A
)  =  ( -u A ^ N ) )
496, 21expp1d 11451 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
( 2  x.  n
)  +  1 ) )  =  ( ( A ^ ( 2  x.  n ) )  x.  A ) )
507oveq2d 6036 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
( 2  x.  n
)  +  1 ) )  =  ( A ^ N ) )
5149, 50eqtr3d 2421 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( A ^
( 2  x.  n
) )  x.  A
)  =  ( A ^ N ) )
5251negeqd 9232 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  -u ( ( A ^
( 2  x.  n
) )  x.  A
)  =  -u ( A ^ N ) )
5348, 52eqtr3d 2421 . 2  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ N
)  =  -u ( A ^ N ) )
545, 53rexlimddv 2777 1  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( -u A ^ N
)  =  -u ( A ^ N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2650   class class class wbr 4153  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223   -ucneg 9224   NNcn 9932   2c2 9981   NN0cn0 10153   ZZcz 10214   ^cexp 11309    || cdivides 12779
This theorem is referenced by:  dcubic1lem  20550  dcubic2  20551  mcubic  20554  lgseisenlem1  21000  lgseisenlem4  21003  m1lgs  21013  stirlinglem5  27495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-seq 11251  df-exp 11310  df-dvds 12780
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