MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oexpneg Structured version   Unicode version

Theorem oexpneg 14133
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 10882 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
2 odd2np1 14130 . . . . 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 482 . . 3  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
543adant1 1012 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N )
6 simpl1 997 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  A  e.  CC )
7 simprr 755 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( 2  x.  n )  +  1 )  =  N )
8 simpl2 998 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  NN )
98nncnd 10547 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  CC )
10 1cnd 9601 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
1  e.  CC )
11 2z 10892 . . . . . . . . . . 11  |-  2  e.  ZZ
12 simprl 754 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  ZZ )
13 zmulcl 10908 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2  x.  n
)  e.  ZZ )
1411, 12, 13sylancr 661 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  ZZ )
1514zcnd 10966 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  CC )
169, 10, 15subadd2d 9941 . . . . . . . 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 ) )
177, 16mpbird 232 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( N  -  1 )  =  ( 2  x.  n ) )
18 nnm1nn0 10833 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
198, 18syl 16 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( N  -  1 )  e.  NN0 )
2017, 19eqeltrrd 2543 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  NN0 )
216, 20expcld 12292 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
2  x.  n ) )  e.  CC )
2221, 6mulneg2d 10006 . . . 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 ) )
23 sqneg 12210 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
246, 23syl 16 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ 2 )  =  ( A ^ 2 ) )
2524oveq1d 6285 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( ( -u A ^ 2 ) ^
n )  =  ( ( A ^ 2 ) ^ n ) )
266negcld 9909 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  -u A  e.  CC )
27 2re 10601 . . . . . . . . . . 11  |-  2  e.  RR
2827a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
2  e.  RR )
2912zred 10965 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  RR )
30 2pos 10623 . . . . . . . . . . 11  |-  0  <  2
3130a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <  2 )
3220nn0ge0d 10851 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  ( 2  x.  n ) )
33 prodge0 10385 . . . . . . . . . 10  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
3428, 29, 31, 32, 33syl22anc 1227 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  n )
35 elnn0z 10873 . . . . . . . . 9  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
3612, 34, 35sylanbrc 662 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  NN0 )
37 2nn0 10808 . . . . . . . . 9  |-  2  e.  NN0
3837a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
2  e.  NN0 )
3926, 36, 38expmuld 12295 . . . . . . 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 ) )
406, 36, 38expmuld 12295 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
2  x.  n ) )  =  ( ( A ^ 2 ) ^ n ) )
4125, 39, 403eqtr4d 2505 . . . . . 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 ) ) )
4241oveq1d 6285 . . . . 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
) )
4326, 20expp1d 12293 . . . . . 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
) )
447oveq2d 6286 . . . . . 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 ) )
4543, 44eqtr3d 2497 . . . . 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
) )
4642, 45eqtr3d 2497 . . . 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 ) )
4722, 46eqtr3d 2497 . . 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 ) )
486, 20expp1d 12293 . . . . 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 ) )
497oveq2d 6286 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( A ^ (
( 2  x.  n
)  +  1 ) )  =  ( A ^ N ) )
5048, 49eqtr3d 2497 . . . 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 ) )
5150negeqd 9805 . . 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 ) )
5247, 51eqtr3d 2497 . 2  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( -u A ^ N
)  =  -u ( A ^ N ) )
535, 52rexlimddv 2950 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805   class class class wbr 4439  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9796   -ucneg 9797   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   ^cexp 12148    || cdvds 14070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12090  df-exp 12149  df-dvds 14071
This theorem is referenced by:  dcubic1lem  23371  dcubic2  23372  mcubic  23375  lgseisenlem1  23822  lgseisenlem4  23825  m1lgs  23835  stirlinglem5  32099
  Copyright terms: Public domain W3C validator