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Theorem m1expevenALT 27250
Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
Assertion
Ref Expression
m1expevenALT  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )

Proof of Theorem m1expevenALT
StepHypRef Expression
1 zcn 10761 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
212timesd 10677 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  N )  =  ( N  +  N ) )
32oveq2d 6215 . 2  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  ( -u 1 ^ ( N  +  N ) ) )
4 neg1cn 10535 . . . 4  |-  -u 1  e.  CC
5 neg1ne0 10537 . . . 4  |-  -u 1  =/=  0
6 expaddz 12024 . . . 4  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  ( N  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( -u 1 ^ ( N  +  N )
)  =  ( (
-u 1 ^ N
)  x.  ( -u
1 ^ N ) ) )
74, 5, 6mpanl12 682 . . 3  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u 1 ^ ( N  +  N
) )  =  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) ) )
87anidms 645 . 2  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( N  +  N ) )  =  ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) ) )
9 m1expcl2 12003 . . 3  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
10 ovex 6224 . . . . 5  |-  ( -u
1 ^ N )  e.  _V
1110elpr 4002 . . . 4  |-  ( (
-u 1 ^ N
)  e.  { -u
1 ,  1 }  <-> 
( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 ) )
12 oveq12 6208 . . . . . . 7  |-  ( ( ( -u 1 ^ N )  =  -u
1  /\  ( -u 1 ^ N )  =  -u
1 )  ->  (
( -u 1 ^ N
)  x.  ( -u
1 ^ N ) )  =  ( -u
1  x.  -u 1
) )
1312anidms 645 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( -u
1  x.  -u 1
) )
14 neg1mulneg1e1 10649 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
1513, 14syl6eq 2511 . . . . 5  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
16 oveq12 6208 . . . . . . 7  |-  ( ( ( -u 1 ^ N )  =  1  /\  ( -u 1 ^ N )  =  1 )  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( 1  x.  1 ) )
1716anidms 645 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( 1  x.  1 ) )
18 1t1e1 10579 . . . . . 6  |-  ( 1  x.  1 )  =  1
1917, 18syl6eq 2511 . . . . 5  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
2015, 19jaoi 379 . . . 4  |-  ( ( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 )  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
2111, 20sylbi 195 . . 3  |-  ( (
-u 1 ^ N
)  e.  { -u
1 ,  1 }  ->  ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
229, 21syl 16 . 2  |-  ( N  e.  ZZ  ->  (
( -u 1 ^ N
)  x.  ( -u
1 ^ N ) )  =  1 )
233, 8, 223eqtrd 2499 1  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   {cpr 3986  (class class class)co 6199   CCcc 9390   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397   -ucneg 9706   2c2 10481   ZZcz 10756   ^cexp 11981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-2nd 6687  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-seq 11923  df-exp 11982
This theorem is referenced by:  fallrisefac  27671
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