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Theorem m1expcl2 11125
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 9050 . . 3  |-  -u 1  e.  _V
21prid1 3734 . 2  |-  -u 1  e.  { -u 1 ,  1 }
3 ax-1cn 8795 . . 3  |-  1  e.  CC
4 ax-1ne0 8806 . . 3  |-  1  =/=  0
53, 4negne0i 9121 . 2  |-  -u 1  =/=  0
6 neg1cn 9813 . . . 4  |-  -u 1  e.  CC
7 prssi 3771 . . . 4  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
86, 3, 7mp2an 653 . . 3  |-  { -u
1 ,  1 } 
C_  CC
9 elpri 3660 . . . . 5  |-  ( x  e.  { -u 1 ,  1 }  ->  ( x  =  -u 1  \/  x  =  1
) )
108sseli 3176 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  CC )
1110mulm1d 9231 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  =  -u y
)
12 elpri 3660 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  ( y  =  -u 1  \/  y  =  1
) )
13 negeq 9044 . . . . . . . . . . 11  |-  ( y  =  -u 1  ->  -u y  =  -u -u 1 )
143negnegi 9116 . . . . . . . . . . . 12  |-  -u -u 1  =  1
15 1ex 8833 . . . . . . . . . . . . 13  |-  1  e.  _V
1615prid2 3735 . . . . . . . . . . . 12  |-  1  e.  { -u 1 ,  1 }
1714, 16eqeltri 2353 . . . . . . . . . . 11  |-  -u -u 1  e.  { -u 1 ,  1 }
1813, 17syl6eqel 2371 . . . . . . . . . 10  |-  ( y  =  -u 1  ->  -u y  e.  { -u 1 ,  1 } )
19 negeq 9044 . . . . . . . . . . 11  |-  ( y  =  1  ->  -u y  =  -u 1 )
2019, 2syl6eqel 2371 . . . . . . . . . 10  |-  ( y  =  1  ->  -u y  e.  { -u 1 ,  1 } )
2118, 20jaoi 368 . . . . . . . . 9  |-  ( ( y  =  -u 1  \/  y  =  1
)  ->  -u y  e. 
{ -u 1 ,  1 } )
2212, 21syl 15 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  -u y  e.  { -u 1 ,  1 } )
2311, 22eqeltrd 2357 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  e.  { -u
1 ,  1 } )
24 oveq1 5865 . . . . . . . 8  |-  ( x  =  -u 1  ->  (
x  x.  y )  =  ( -u 1  x.  y ) )
2524eleq1d 2349 . . . . . . 7  |-  ( x  =  -u 1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( -u 1  x.  y
)  e.  { -u
1 ,  1 } ) )
2623, 25syl5ibr 212 . . . . . 6  |-  ( x  =  -u 1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
2710mulid2d 8853 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  =  y )
28 id 19 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  { -u 1 ,  1 } )
2927, 28eqeltrd 2357 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  e.  { -u 1 ,  1 } )
30 oveq1 5865 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
3130eleq1d 2349 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( 1  x.  y
)  e.  { -u
1 ,  1 } ) )
3229, 31syl5ibr 212 . . . . . 6  |-  ( x  =  1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3326, 32jaoi 368 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( y  e.  { -u 1 ,  1 }  ->  (
x  x.  y )  e.  { -u 1 ,  1 } ) )
349, 33syl 15 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3534imp 418 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  y  e.  { -u 1 ,  1 } )  ->  ( x  x.  y )  e.  { -u 1 ,  1 } )
36 oveq2 5866 . . . . . . 7  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
37 divneg2 9484 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
383, 3, 4, 37mp3an 1277 . . . . . . . . 9  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
393div1i 9488 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4039negeqi 9045 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
4138, 40eqtr3i 2305 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
4241, 2eqeltri 2353 . . . . . . 7  |-  ( 1  /  -u 1 )  e. 
{ -u 1 ,  1 }
4336, 42syl6eqel 2371 . . . . . 6  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
44 oveq2 5866 . . . . . . 7  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
4539, 16eqeltri 2353 . . . . . . 7  |-  ( 1  /  1 )  e. 
{ -u 1 ,  1 }
4644, 45syl6eqel 2371 . . . . . 6  |-  ( x  =  1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
4743, 46jaoi 368 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
489, 47syl 15 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( 1  /  x )  e.  { -u 1 ,  1 } )
4948adantr 451 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  x  =/=  0
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
508, 35, 16, 49expcl2lem 11115 . 2  |-  ( (
-u 1  e.  { -u 1 ,  1 }  /\  -u 1  =/=  0  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
512, 5, 50mp3an12 1267 1  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   {cpr 3641  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   -ucneg 9038    / cdiv 9423   ZZcz 10024   ^cexp 11104
This theorem is referenced by:  m1expcl  11126  lgseisenlem2  20589  m1expaddsub  27421  psgnghm  27437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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