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Theorem m1expcl2 12167
Description: Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
Assertion
Ref Expression
m1expcl2  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )

Proof of Theorem m1expcl2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negex 9823 . . 3  |-  -u 1  e.  _V
21prid1 4123 . 2  |-  -u 1  e.  { -u 1 ,  1 }
3 neg1ne0 10647 . 2  |-  -u 1  =/=  0
4 neg1cn 10645 . . . 4  |-  -u 1  e.  CC
5 ax-1cn 9553 . . . 4  |-  1  e.  CC
6 prssi 4171 . . . 4  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
74, 5, 6mp2an 672 . . 3  |-  { -u
1 ,  1 } 
C_  CC
8 elpri 4034 . . . . 5  |-  ( x  e.  { -u 1 ,  1 }  ->  ( x  =  -u 1  \/  x  =  1
) )
97sseli 3485 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  CC )
109mulm1d 10014 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  =  -u y
)
11 elpri 4034 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  ( y  =  -u 1  \/  y  =  1
) )
12 negeq 9817 . . . . . . . . . . 11  |-  ( y  =  -u 1  ->  -u y  =  -u -u 1 )
13 negneg1e1 10649 . . . . . . . . . . . 12  |-  -u -u 1  =  1
14 1ex 9594 . . . . . . . . . . . . 13  |-  1  e.  _V
1514prid2 4124 . . . . . . . . . . . 12  |-  1  e.  { -u 1 ,  1 }
1613, 15eqeltri 2527 . . . . . . . . . . 11  |-  -u -u 1  e.  { -u 1 ,  1 }
1712, 16syl6eqel 2539 . . . . . . . . . 10  |-  ( y  =  -u 1  ->  -u y  e.  { -u 1 ,  1 } )
18 negeq 9817 . . . . . . . . . . 11  |-  ( y  =  1  ->  -u y  =  -u 1 )
1918, 2syl6eqel 2539 . . . . . . . . . 10  |-  ( y  =  1  ->  -u y  e.  { -u 1 ,  1 } )
2017, 19jaoi 379 . . . . . . . . 9  |-  ( ( y  =  -u 1  \/  y  =  1
)  ->  -u y  e. 
{ -u 1 ,  1 } )
2111, 20syl 16 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  -u y  e.  { -u 1 ,  1 } )
2210, 21eqeltrd 2531 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  e.  { -u
1 ,  1 } )
23 oveq1 6288 . . . . . . . 8  |-  ( x  =  -u 1  ->  (
x  x.  y )  =  ( -u 1  x.  y ) )
2423eleq1d 2512 . . . . . . 7  |-  ( x  =  -u 1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( -u 1  x.  y
)  e.  { -u
1 ,  1 } ) )
2522, 24syl5ibr 221 . . . . . 6  |-  ( x  =  -u 1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
269mulid2d 9617 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  =  y )
27 id 22 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  { -u 1 ,  1 } )
2826, 27eqeltrd 2531 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  e.  { -u 1 ,  1 } )
29 oveq1 6288 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
3029eleq1d 2512 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( 1  x.  y
)  e.  { -u
1 ,  1 } ) )
3128, 30syl5ibr 221 . . . . . 6  |-  ( x  =  1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3225, 31jaoi 379 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( y  e.  { -u 1 ,  1 }  ->  (
x  x.  y )  e.  { -u 1 ,  1 } ) )
338, 32syl 16 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3433imp 429 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  y  e.  { -u 1 ,  1 } )  ->  ( x  x.  y )  e.  { -u 1 ,  1 } )
35 oveq2 6289 . . . . . . 7  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
36 ax-1ne0 9564 . . . . . . . . . 10  |-  1  =/=  0
37 divneg2 10274 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
385, 5, 36, 37mp3an 1325 . . . . . . . . 9  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
39 1div1e1 10243 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4039negeqi 9818 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
4138, 40eqtr3i 2474 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
4241, 2eqeltri 2527 . . . . . . 7  |-  ( 1  /  -u 1 )  e. 
{ -u 1 ,  1 }
4335, 42syl6eqel 2539 . . . . . 6  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
44 oveq2 6289 . . . . . . 7  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
4539, 15eqeltri 2527 . . . . . . 7  |-  ( 1  /  1 )  e. 
{ -u 1 ,  1 }
4644, 45syl6eqel 2539 . . . . . 6  |-  ( x  =  1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
4743, 46jaoi 379 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
488, 47syl 16 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( 1  /  x )  e.  { -u 1 ,  1 } )
4948adantr 465 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  x  =/=  0
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
507, 34, 15, 49expcl2lem 12157 . 2  |-  ( (
-u 1  e.  { -u 1 ,  1 }  /\  -u 1  =/=  0  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
512, 3, 50mp3an12 1315 1  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1383    e. wcel 1804    =/= wne 2638    C_ wss 3461   {cpr 4016  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    x. cmul 9500   -ucneg 9811    / cdiv 10212   ZZcz 10870   ^cexp 12145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-n0 10802  df-z 10871  df-uz 11091  df-seq 12087  df-exp 12146
This theorem is referenced by:  m1expcl  12168  m1expaddsub  16397  psgnran  16414  psgnghm  18489  lgseisenlem2  23497  m1expevenALT  28536
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