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Theorem oexpneg 12606
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 10061 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
2 odd2np1 12603 . . . . 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 9778 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  N  e.  CC )
10 ax-1cn 8811 . . . . . . . . . . . 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 10070 . . . . . . . . . . . . 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 10082 . . . . . . . . . . . . 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 10134 . . . . . . . . . . 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 9192 . . . . . . . . . 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 10021 . . . . . . . . . 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 2371 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
( 2  x.  n
)  e.  NN0 )
226, 21expcld 11261 . . . . . . 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 9249 . . . . . 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 11180 . . . . . . . . . . 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 5889 . . . . . . . . 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 9160 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  -u A  e.  CC )
28 2re 9831 . . . . . . . . . . . . 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 10133 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  ->  n  e.  RR )
31 2pos 9844 . . . . . . . . . . . . 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 10037 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  NN  /\  -.  2  ||  N )  /\  ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N ) )  -> 
0  <_  ( 2  x.  n ) )
34 prodge0 9619 . . . . . . . . . . . 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 10052 . . . . . . . . . . 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 9998 . . . . . . . . . . 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 11264 . . . . . . . . 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 11264 . . . . . . . . 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 2338 . . . . . . . 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 5889 . . . . . . 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 11262 . . . . . . . 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 5890 . . . . . . . 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 2330 . . . . . . 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 2330 . . . . . 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 2330 . . . . 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 11262 . . . . . . 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 5890 . . . . . . 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 2330 . . . . . 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 9062 . . . . 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 2330 . . . 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 2680 . 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 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ^cexp 11120    || cdivides 12547
This theorem is referenced by:  dcubic1lem  20155  dcubic2  20156  mcubic  20159  lgseisenlem1  20604  lgseisenlem4  20607  m1lgs  20617  stirlinglem5  27930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121  df-dvds 12548
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