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Theorem oexpneg 13593
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 10666 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
2 odd2np1 13590 . . . . 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 484 . . 3  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
543adant1 1006 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
6 simpl1 991 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  A  e.  CC )
7 simprr 756 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( 2  x.  n )  +  1 )  =  N )
8 simpl2 992 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  NN )
98nncnd 10336 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  CC )
10 ax-1cn 9338 . . . . . . . . . 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 10676 . . . . . . . . . . 11  |-  2  e.  ZZ
13 simprl 755 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  ZZ )
14 zmulcl 10691 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2  x.  n
)  e.  ZZ )
1512, 13, 14sylancr 663 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  ZZ )
1615zcnd 10746 . . . . . . . . 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 9736 . . . . . . . 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 232 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( N  -  1 )  =  ( 2  x.  n ) )
19 nnm1nn0 10619 . . . . . . . 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 2516 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  NN0 )
226, 21expcld 12006 . . . . 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 9796 . . . 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 11924 . . . . . . . . 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 6104 . . . . . . 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 9704 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  -u A  e.  CC )
28 2re 10389 . . . . . . . . . . 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 10745 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  RR )
31 2pos 10411 . . . . . . . . . . 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 10637 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  ( 2  x.  n ) )
34 prodge0 10174 . . . . . . . . . 10  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
3529, 30, 32, 33, 34syl22anc 1219 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  n )
36 elnn0z 10657 . . . . . . . . 9  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
3713, 35, 36sylanbrc 664 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  NN0 )
38 2nn0 10594 . . . . . . . . 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 12009 . . . . . . 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 12009 . . . . . . 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 2483 . . . . . 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 6104 . . . . 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 12007 . . . . . 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 6105 . . . . . 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 2475 . . . . 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 2475 . . . 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 2475 . . 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 12007 . . . . 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 6105 . . . . 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 2475 . . . 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 9602 . . 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 2475 . 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 2843 1  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( -u A ^ N
)  =  -u ( A ^ N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2714   class class class wbr 4290  (class class class)co 6089   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    + caddc 9283    x. cmul 9285    < clt 9416    <_ cle 9417    - cmin 9593   -ucneg 9594   NNcn 10320   2c2 10369   NN0cn0 10577   ZZcz 10644   ^cexp 11863    || cdivides 13533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-seq 11805  df-exp 11864  df-dvds 13534
This theorem is referenced by:  dcubic1lem  22236  dcubic2  22237  mcubic  22240  lgseisenlem1  22686  lgseisenlem4  22689  m1lgs  22699  stirlinglem5  29870
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