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Theorem m1exp1 10508
Description: Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)
Assertion
Ref Expression
m1exp1  |-  ( N  e.  ZZ  ->  (
( -u 1 ^ N
)  =  1  <->  2 
||  N ) )

Proof of Theorem m1exp1
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 2z 8512 . . . . . . 7  |-  2  e.  ZZ
2 divides 10405 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ )  ->  ( 2  ||  N  <->  E. n  e.  ZZ  (
n  x.  2 )  =  N ) )
31, 2mpan 415 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  ||  N  <->  E. n  e.  ZZ  ( n  x.  2 )  =  N ) )
4 oveq2 5571 . . . . . . . . 9  |-  ( N  =  ( n  x.  2 )  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( n  x.  2 ) ) )
54eqcoms 2086 . . . . . . . 8  |-  ( ( n  x.  2 )  =  N  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( n  x.  2 ) ) )
6 zcn 8489 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  n  e.  CC )
7 2cnd 8231 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  2  e.  CC )
86, 7mulcomd 7254 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  (
n  x.  2 )  =  ( 2  x.  n ) )
98oveq2d 5579 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( n  x.  2 ) )  =  ( -u 1 ^ ( 2  x.  n ) ) )
10 m1expeven 9672 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  n ) )  =  1 )
119, 10eqtrd 2115 . . . . . . . 8  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( n  x.  2 ) )  =  1 )
125, 11sylan9eqr 2137 . . . . . . 7  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  N )  ->  ( -u 1 ^ N )  =  1 )
1312rexlimiva 2477 . . . . . 6  |-  ( E. n  e.  ZZ  (
n  x.  2 )  =  N  ->  ( -u 1 ^ N )  =  1 )
143, 13syl6bi 161 . . . . 5  |-  ( N  e.  ZZ  ->  (
2  ||  N  ->  (
-u 1 ^ N
)  =  1 ) )
1514impcom 123 . . . 4  |-  ( ( 2  ||  N  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  =  1 )
16 simpl 107 . . . 4  |-  ( ( 2  ||  N  /\  N  e.  ZZ )  ->  2  ||  N )
1715, 162thd 173 . . 3  |-  ( ( 2  ||  N  /\  N  e.  ZZ )  ->  ( ( -u 1 ^ N )  =  1  <->  2  ||  N ) )
1817expcom 114 . 2  |-  ( N  e.  ZZ  ->  (
2  ||  N  ->  ( ( -u 1 ^ N )  =  1  <->  2  ||  N ) ) )
19 1ne0 8226 . . . . . 6  |-  1  =/=  0
20 eqcom 2085 . . . . . . 7  |-  ( -u
1  =  1  <->  1  =  -u 1 )
21 ax-1cn 7183 . . . . . . . 8  |-  1  e.  CC
2221eqnegi 7948 . . . . . . 7  |-  ( 1  =  -u 1  <->  1  = 
0 )
2320, 22bitri 182 . . . . . 6  |-  ( -u
1  =  1  <->  1  =  0 )
2419, 23nemtbir 2338 . . . . 5  |-  -.  -u 1  =  1
25 odd2np1 10480 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
26 oveq2 5571 . . . . . . . . . . 11  |-  ( N  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) ) )
2726eqcoms 2086 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) ) )
28 neg1cn 8263 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
2928a1i 9 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  -u 1  e.  CC )
30 neg1ap0 8267 . . . . . . . . . . . . 13  |-  -u 1 #  0
3130a1i 9 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  -u 1 #  0 )
321a1i 9 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  2  e.  ZZ )
33 id 19 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  ZZ )
3432, 33zmulcld 8608 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
2  x.  n )  e.  ZZ )
3529, 31, 34expp1zapd 9763 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) )  =  ( ( -u
1 ^ ( 2  x.  n ) )  x.  -u 1 ) )
3610oveq1d 5578 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
( -u 1 ^ (
2  x.  n ) )  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
3728mulid2i 7236 . . . . . . . . . . . 12  |-  ( 1  x.  -u 1 )  = 
-u 1
3836, 37syl6eq 2131 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
( -u 1 ^ (
2  x.  n ) )  x.  -u 1
)  =  -u 1
)
3935, 38eqtrd 2115 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) )  =  -u 1 )
4027, 39sylan9eqr 2137 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N )  ->  ( -u 1 ^ N )  =  -u
1 )
4140rexlimiva 2477 . . . . . . . 8  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  N  ->  ( -u 1 ^ N )  =  -u 1 )
4225, 41syl6bi 161 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  -> 
( -u 1 ^ N
)  =  -u 1
) )
4342impcom 123 . . . . . 6  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  =  -u
1 )
4443eqeq1d 2091 . . . . 5  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  ( ( -u
1 ^ N )  =  1  <->  -u 1  =  1 ) )
4524, 44mtbiri 633 . . . 4  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  -.  ( -u 1 ^ N )  =  1 )
46 simpl 107 . . . 4  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  -.  2  ||  N )
4745, 462falsed 651 . . 3  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  ( ( -u
1 ^ N )  =  1  <->  2  ||  N ) )
4847expcom 114 . 2  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  -> 
( ( -u 1 ^ N )  =  1  <->  2  ||  N ) ) )
49 zeo3 10475 . 2  |-  ( N  e.  ZZ  ->  (
2  ||  N  \/  -.  2  ||  N ) )
5018, 48, 49mpjaod 671 1  |-  ( N  e.  ZZ  ->  (
( -u 1 ^ N
)  =  1  <->  2 
||  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2354   class class class wbr 3805  (class class class)co 5563   CCcc 7093   0cc0 7095   1c1 7096    + caddc 7098    x. cmul 7100   -ucneg 7399   # cap 7800   2c2 8208   ZZcz 8484   ^cexp 9624    || cdvds 10403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-xor 1308  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-n0 8408  df-z 8485  df-uz 8753  df-iseq 9574  df-iexp 9625  df-dvds 10404
This theorem is referenced by: (None)
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