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Theorem m1expeven 9620
Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
Assertion
Ref Expression
m1expeven  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )

Proof of Theorem m1expeven
StepHypRef Expression
1 zcn 8437 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
212timesd 8340 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  N )  =  ( N  +  N ) )
32oveq2d 5559 . 2  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  ( -u 1 ^ ( N  +  N ) ) )
4 neg1cn 8211 . . . 4  |-  -u 1  e.  CC
5 neg1ap0 8215 . . . 4  |-  -u 1 #  0
6 expaddzap 9617 . . . 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 427 . . 3  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u 1 ^ ( N  +  N
) )  =  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) ) )
87anidms 389 . 2  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( N  +  N ) )  =  ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) ) )
9 m1expcl2 9595 . . 3  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
10 neg1rr 8212 . . . . . 6  |-  -u 1  e.  RR
11 reexpclzap 9593 . . . . . 6  |-  ( (
-u 1  e.  RR  /\  -u 1 #  0  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  e.  RR )
1210, 5, 11mp3an12 1259 . . . . 5  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  RR )
13 elprg 3426 . . . . 5  |-  ( (
-u 1 ^ N
)  e.  RR  ->  ( ( -u 1 ^ N )  e.  { -u 1 ,  1 }  <-> 
( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 ) ) )
1412, 13syl 14 . . . 4  |-  ( N  e.  ZZ  ->  (
( -u 1 ^ N
)  e.  { -u
1 ,  1 }  <-> 
( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 ) ) )
15 oveq12 5552 . . . . . . 7  |-  ( ( ( -u 1 ^ N )  =  -u
1  /\  ( -u 1 ^ N )  =  -u
1 )  ->  (
( -u 1 ^ N
)  x.  ( -u
1 ^ N ) )  =  ( -u
1  x.  -u 1
) )
1615anidms 389 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( -u
1  x.  -u 1
) )
17 neg1mulneg1e1 8310 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
1816, 17syl6eq 2130 . . . . 5  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
19 oveq12 5552 . . . . . . 7  |-  ( ( ( -u 1 ^ N )  =  1  /\  ( -u 1 ^ N )  =  1 )  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( 1  x.  1 ) )
2019anidms 389 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  ( 1  x.  1 ) )
21 1t1e1 8251 . . . . . 6  |-  ( 1  x.  1 )  =  1
2220, 21syl6eq 2130 . . . . 5  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
2318, 22jaoi 669 . . . 4  |-  ( ( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 )  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
2414, 23syl6bi 161 . . 3  |-  ( N  e.  ZZ  ->  (
( -u 1 ^ N
)  e.  { -u
1 ,  1 }  ->  ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 ) )
259, 24mpd 13 . 2  |-  ( N  e.  ZZ  ->  (
( -u 1 ^ N
)  x.  ( -u
1 ^ N ) )  =  1 )
263, 8, 253eqtrd 2118 1  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   {cpr 3407   class class class wbr 3793  (class class class)co 5543   CCcc 7041   RRcr 7042   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048   -ucneg 7347   # cap 7748   2c2 8156   ZZcz 8432   ^cexp 9572
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522  df-iexp 9573
This theorem is referenced by:  m1expe  10443  m1expo  10444  m1exp1  10445
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