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Theorem m1expevenALT 28927
Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
m1expevenALT  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )

Proof of Theorem m1expevenALT
StepHypRef Expression
1 zcn 10865 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
212timesd 10777 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  N )  =  ( N  +  N ) )
32oveq2d 6286 . 2  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  ( -u 1 ^ ( N  +  N ) ) )
4 neg1cn 10635 . . . 4  |-  -u 1  e.  CC
5 neg1ne0 10637 . . . 4  |-  -u 1  =/=  0
6 expaddz 12192 . . . 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 680 . . 3  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u 1 ^ ( N  +  N
) )  =  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) ) )
87anidms 643 . 2  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( N  +  N ) )  =  ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) ) )
9 m1expcl2 12170 . . 3  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
10 ovex 6298 . . . . 5  |-  ( -u
1 ^ N )  e.  _V
1110elpr 4034 . . . 4  |-  ( (
-u 1 ^ N
)  e.  { -u
1 ,  1 }  <-> 
( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 ) )
12 oveq12 6279 . . . . . . 7  |-  ( ( ( -u 1 ^ N )  =  -u
1  /\  ( -u 1 ^ N )  =  -u
1 )  ->  (
( -u 1 ^ N
)  x.  ( -u
1 ^ N ) )  =  ( -u
1  x.  -u 1
) )
1312anidms 643 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( -u
1  x.  -u 1
) )
14 neg1mulneg1e1 10749 . . . . . 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 6279 . . . . . . 7  |-  ( ( ( -u 1 ^ N )  =  1  /\  ( -u 1 ^ N )  =  1 )  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( 1  x.  1 ) )
1716anidms 643 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( 1  x.  1 ) )
18 1t1e1 10679 . . . . . 6  |-  ( 1  x.  1 )  =  1
1917, 18syl6eq 2511 . . . . 5  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
2015, 19jaoi 377 . . . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {cpr 4018  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   -ucneg 9797   2c2 10581   ZZcz 10860   ^cexp 12148
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
This theorem is referenced by:  fallrisefac  29388
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