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Theorem m1exp1 11505
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 9036 . . . . . . 7  |-  2  e.  ZZ
2 divides 11402 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ )  ->  ( 2  ||  N  <->  E. n  e.  ZZ  (
n  x.  2 )  =  N ) )
31, 2mpan 418 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  ||  N  <->  E. n  e.  ZZ  ( n  x.  2 )  =  N ) )
4 oveq2 5748 . . . . . . . . 9  |-  ( N  =  ( n  x.  2 )  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( n  x.  2 ) ) )
54eqcoms 2118 . . . . . . . 8  |-  ( ( n  x.  2 )  =  N  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( n  x.  2 ) ) )
6 zcn 9013 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  n  e.  CC )
7 2cnd 8753 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  2  e.  CC )
86, 7mulcomd 7751 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  (
n  x.  2 )  =  ( 2  x.  n ) )
98oveq2d 5756 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( n  x.  2 ) )  =  ( -u 1 ^ ( 2  x.  n ) ) )
10 m1expeven 10291 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  n ) )  =  1 )
119, 10eqtrd 2148 . . . . . . . 8  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( n  x.  2 ) )  =  1 )
125, 11sylan9eqr 2170 . . . . . . 7  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  N )  ->  ( -u 1 ^ N )  =  1 )
1312rexlimiva 2519 . . . . . 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 8748 . . . . . 6  |-  1  =/=  0
20 eqcom 2117 . . . . . . 7  |-  ( -u
1  =  1  <->  1  =  -u 1 )
21 ax-1cn 7677 . . . . . . . 8  |-  1  e.  CC
2221eqnegi 8464 . . . . . . 7  |-  ( 1  =  -u 1  <->  1  = 
0 )
2320, 22bitri 183 . . . . . 6  |-  ( -u
1  =  1  <->  1  =  0 )
2419, 23nemtbir 2372 . . . . 5  |-  -.  -u 1  =  1
25 odd2np1 11477 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( -.  2  ||  N  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
26 oveq2 5748 . . . . . . . . . . 11  |-  ( N  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) ) )
2726eqcoms 2118 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  ( -u 1 ^ N )  =  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) ) )
28 neg1cn 8785 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
2928a1i 9 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  -u 1  e.  CC )
30 neg1ap0 8789 . . . . . . . . . . . . 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 9133 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
2  x.  n )  e.  ZZ )
3529, 31, 34expp1zapd 10384 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) )  =  ( ( -u
1 ^ ( 2  x.  n ) )  x.  -u 1 ) )
3610oveq1d 5755 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
( -u 1 ^ (
2  x.  n ) )  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
3728mulid2i 7733 . . . . . . . . . . . 12  |-  ( 1  x.  -u 1 )  = 
-u 1
3836, 37syl6eq 2164 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
( -u 1 ^ (
2  x.  n ) )  x.  -u 1
)  =  -u 1
)
3935, 38eqtrd 2148 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  ( -u 1 ^ ( ( 2  x.  n )  +  1 ) )  =  -u 1 )
4027, 39sylan9eqr 2170 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  N )  ->  ( -u 1 ^ N )  =  -u
1 )
4140rexlimiva 2519 . . . . . . . 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 2124 . . . . 5  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  ( ( -u
1 ^ N )  =  1  <->  -u 1  =  1 ) )
4524, 44mtbiri 647 . . . 4  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  -.  ( -u 1 ^ N )  =  1 )
46 simpl 108 . . . 4  |-  ( ( -.  2  ||  N  /\  N  e.  ZZ )  ->  -.  2  ||  N )
4745, 462falsed 674 . . 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 11472 . 2  |-  ( N  e.  ZZ  ->  (
2  ||  N  \/  -.  2  ||  N ) )
5018, 48, 49mpjaod 690 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 1314    e. wcel 1463   E.wrex 2392   class class class wbr 3897  (class class class)co 5740   CCcc 7582   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589   -ucneg 7898   # cap 8306   2c2 8731   ZZcz 9008   ^cexp 10243    || cdvds 11400
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-xor 1337  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-n0 8932  df-z 9009  df-uz 9279  df-seqfrec 10170  df-exp 10244  df-dvds 11401
This theorem is referenced by: (None)
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