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Theorem oexpneg 12590
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 10045 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
2 odd2np1 12587 . . . . 5  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
31, 2syl 15 . . . 4  |-  ( N  e.  NN  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
43biimpa 470 . . 3  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
543adant1 973 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
6 simpl1 958 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  A  e.  CC )
7 simprr 733 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( 2  x.  n )  +  1 )  =  N )
8 simpl2 959 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  NN )
98nncnd 9762 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  CC )
10 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
1110a1i 10 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
1  e.  CC )
12 2z 10054 . . . . . . . . . . . . 13  |-  2  e.  ZZ
13 simprl 732 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  ZZ )
14 zmulcl 10066 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2  x.  n
)  e.  ZZ )
1512, 13, 14sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  ZZ )
1615zcnd 10118 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  CC )
179, 11, 16subadd2d 9176 . . . . . . . . . 10  |-  ( ( ( 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 223 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( N  -  1 )  =  ( 2  x.  n ) )
19 nnm1nn0 10005 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
208, 19syl 15 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( N  -  1 )  e.  NN0 )
2118, 20eqeltrrd 2358 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  NN0 )
226, 21expcld 11245 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
2  x.  n ) )  e.  CC )
2322, 6mulneg2d 9233 . . . . . 6  |-  ( ( ( 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 11164 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
256, 24syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ 2 )  =  ( A ^ 2 ) )
2625oveq1d 5873 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( -u A ^ 2 ) ^
n )  =  ( ( A ^ 2 ) ^ n ) )
276negcld 9144 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  -u A  e.  CC )
28 2re 9815 . . . . . . . . . . . . 13  |-  2  e.  RR
2928a1i 10 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
2  e.  RR )
3013zred 10117 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  RR )
31 2pos 9828 . . . . . . . . . . . . 13  |-  0  <  2
3231a1i 10 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <  2 )
3321nn0ge0d 10021 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  ( 2  x.  n ) )
34 prodge0 9603 . . . . . . . . . . . 12  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
3529, 30, 32, 33, 34syl22anc 1183 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  n )
36 elnn0z 10036 . . . . . . . . . . 11  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
3713, 35, 36sylanbrc 645 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  NN0 )
38 2nn0 9982 . . . . . . . . . . 11  |-  2  e.  NN0
3938a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
2  e.  NN0 )
4027, 37, 39expmuld 11248 . . . . . . . . 9  |-  ( ( ( 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 11248 . . . . . . . . 9  |-  ( ( ( 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 2325 . . . . . . . 8  |-  ( ( ( 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 5873 . . . . . . 7  |-  ( ( ( 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 11246 . . . . . . . 8  |-  ( ( ( 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 5874 . . . . . . . 8  |-  ( ( ( 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 2317 . . . . . . 7  |-  ( ( ( 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 2317 . . . . . 6  |-  ( ( ( 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 2317 . . . . 5  |-  ( ( ( 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 11246 . . . . . . 7  |-  ( ( ( 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 5874 . . . . . . 7  |-  ( ( ( 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 2317 . . . . . 6  |-  ( ( ( 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 9046 . . . . 5  |-  ( ( ( 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 2317 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ N
)  =  -u ( A ^ N ) )
5453expr 598 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  n  e.  ZZ )  ->  ( ( ( 2  x.  n )  +  1 )  =  N  ->  ( -u A ^ N )  =  -u ( A ^ N ) ) )
5554rexlimdva 2667 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N  ->  ( -u A ^ N )  =  -u ( A ^ N ) ) )
565, 55mpd 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ^cexp 11104    || cdivides 12531
This theorem is referenced by:  dcubic1lem  20139  dcubic2  20140  mcubic  20143  lgseisenlem1  20588  lgseisenlem4  20591  m1lgs  20601  stirlinglem5  27827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105  df-dvds 12532
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