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Theorem m1exp1 11860
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 9240 . . . . . . 7  |-  2  e.  ZZ
2 divides 11751 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ )  ->  ( 2  ||  N  <->  E. n  e.  ZZ  (
n  x.  2 )  =  N ) )
31, 2mpan 422 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  ||  N  <->  E. n  e.  ZZ  ( n  x.  2 )  =  N ) )
4 oveq2 5861 . . . . . . . . 9  |-  ( N  =  ( n  x.  2 )  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( n  x.  2 ) ) )
54eqcoms 2173 . . . . . . . 8  |-  ( ( n  x.  2 )  =  N  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( n  x.  2 ) ) )
6 zcn 9217 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  n  e.  CC )
7 2cnd 8951 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  2  e.  CC )
86, 7mulcomd 7941 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  (
n  x.  2 )  =  ( 2  x.  n ) )
98oveq2d 5869 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( n  x.  2 ) )  =  ( -u 1 ^ ( 2  x.  n ) ) )
10 m1expeven 10523 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  n ) )  =  1 )
119, 10eqtrd 2203 . . . . . . . 8  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( n  x.  2 ) )  =  1 )
125, 11sylan9eqr 2225 . . . . . . 7  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  N )  ->  ( -u 1 ^ N )  =  1 )
1312rexlimiva 2582 . . . . . 6  |-  ( E. n  e.  ZZ  (
n  x.  2 )  =  N  ->  ( -u 1 ^ N )  =  1 )
143, 13syl6bi 162 . . . . 5  |-  ( N  e.  ZZ  ->  (
2  ||  N  ->  (
-u 1 ^ N
)  =  1 ) )
1514impcom 124 . . . 4  |-  ( ( 2  ||  N  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  =  1 )
16 simpl 108 . . . 4  |-  ( ( 2  ||  N  /\  N  e.  ZZ )  ->  2  ||  N )
1715, 162thd 174 . . 3  |-  ( ( 2  ||  N  /\  N  e.  ZZ )  ->  ( ( -u 1 ^ N )  =  1  <->  2  ||  N ) )
1817expcom 115 . 2  |-  ( N  e.  ZZ  ->  (
2  ||  N  ->  ( ( -u 1 ^ N )  =  1  <->  2  ||  N ) ) )
19 1ne0 8946 . . . . . 6  |-  1  =/=  0
20 eqcom 2172 . . . . . . 7  |-  ( -u
1  =  1  <->  1  =  -u 1 )
21 ax-1cn 7867 . . . . . . . 8  |-  1  e.  CC
2221eqnegi 8658 . . . . . . 7  |-  ( 1  =  -u 1  <->  1  = 
0 )
2320, 22bitri 183 . . . . . 6  |-  ( -u
1  =  1  <->  1  =  0 )
2419, 23nemtbir 2429 . . . . 5  |-  -.  -u 1  =  1
25 odd2np1 11832 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
26 oveq2 5861 . . . . . . . . . . 11  |-  ( N  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) ) )
2726eqcoms 2173 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) ) )
28 neg1cn 8983 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
2928a1i 9 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  -u 1  e.  CC )
30 neg1ap0 8987 . . . . . . . . . . . . 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 9340 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
2  x.  n )  e.  ZZ )
3529, 31, 34expp1zapd 10618 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) )  =  ( ( -u
1 ^ ( 2  x.  n ) )  x.  -u 1 ) )
3610oveq1d 5868 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
( -u 1 ^ (
2  x.  n ) )  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
3728mulid2i 7923 . . . . . . . . . . . 12  |-  ( 1  x.  -u 1 )  = 
-u 1
3836, 37eqtrdi 2219 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
( -u 1 ^ (
2  x.  n ) )  x.  -u 1
)  =  -u 1
)
3935, 38eqtrd 2203 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) )  =  -u 1 )
4027, 39sylan9eqr 2225 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N )  ->  ( -u 1 ^ N )  =  -u
1 )
4140rexlimiva 2582 . . . . . . . 8  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  N  ->  ( -u 1 ^ N )  =  -u 1 )
4225, 41syl6bi 162 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  -> 
( -u 1 ^ N
)  =  -u 1
) )
4342impcom 124 . . . . . 6  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  =  -u
1 )
4443eqeq1d 2179 . . . . 5  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  ( ( -u
1 ^ N )  =  1  <->  -u 1  =  1 ) )
4524, 44mtbiri 670 . . . 4  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  -.  ( -u 1 ^ N )  =  1 )
46 simpl 108 . . . 4  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  -.  2  ||  N )
4745, 462falsed 697 . . 3  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  ( ( -u
1 ^ N )  =  1  <->  2  ||  N ) )
4847expcom 115 . 2  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  -> 
( ( -u 1 ^ N )  =  1  <->  2  ||  N ) ) )
49 zeo3 11827 . 2  |-  ( N  e.  ZZ  ->  (
2  ||  N  \/  -.  2  ||  N ) )
5018, 48, 49mpjaod 713 1  |-  ( N  e.  ZZ  ->  (
( -u 1 ^ N
)  =  1  <->  2 
||  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775    + caddc 7777    x. cmul 7779   -ucneg 8091   # cap 8500   2c2 8929   ZZcz 9212   ^cexp 10475    || cdvds 11749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476  df-dvds 11750
This theorem is referenced by: (None)
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