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Theorem m1expcl2 11403
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 9304 . . 3  |-  -u 1  e.  _V
21prid1 3912 . 2  |-  -u 1  e.  { -u 1 ,  1 }
3 ax-1cn 9048 . . 3  |-  1  e.  CC
4 ax-1ne0 9059 . . 3  |-  1  =/=  0
53, 4negne0i 9375 . 2  |-  -u 1  =/=  0
6 neg1cn 10067 . . . 4  |-  -u 1  e.  CC
7 prssi 3954 . . . 4  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
86, 3, 7mp2an 654 . . 3  |-  { -u
1 ,  1 } 
C_  CC
9 elpri 3834 . . . . 5  |-  ( x  e.  { -u 1 ,  1 }  ->  ( x  =  -u 1  \/  x  =  1
) )
108sseli 3344 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  CC )
1110mulm1d 9485 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  =  -u y
)
12 elpri 3834 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  ( y  =  -u 1  \/  y  =  1
) )
13 negeq 9298 . . . . . . . . . . 11  |-  ( y  =  -u 1  ->  -u y  =  -u -u 1 )
143negnegi 9370 . . . . . . . . . . . 12  |-  -u -u 1  =  1
15 1ex 9086 . . . . . . . . . . . . 13  |-  1  e.  _V
1615prid2 3913 . . . . . . . . . . . 12  |-  1  e.  { -u 1 ,  1 }
1714, 16eqeltri 2506 . . . . . . . . . . 11  |-  -u -u 1  e.  { -u 1 ,  1 }
1813, 17syl6eqel 2524 . . . . . . . . . 10  |-  ( y  =  -u 1  ->  -u y  e.  { -u 1 ,  1 } )
19 negeq 9298 . . . . . . . . . . 11  |-  ( y  =  1  ->  -u y  =  -u 1 )
2019, 2syl6eqel 2524 . . . . . . . . . 10  |-  ( y  =  1  ->  -u y  e.  { -u 1 ,  1 } )
2118, 20jaoi 369 . . . . . . . . 9  |-  ( ( y  =  -u 1  \/  y  =  1
)  ->  -u y  e. 
{ -u 1 ,  1 } )
2212, 21syl 16 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  -u y  e.  { -u 1 ,  1 } )
2311, 22eqeltrd 2510 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  e.  { -u
1 ,  1 } )
24 oveq1 6088 . . . . . . . 8  |-  ( x  =  -u 1  ->  (
x  x.  y )  =  ( -u 1  x.  y ) )
2524eleq1d 2502 . . . . . . 7  |-  ( x  =  -u 1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( -u 1  x.  y
)  e.  { -u
1 ,  1 } ) )
2623, 25syl5ibr 213 . . . . . 6  |-  ( x  =  -u 1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
2710mulid2d 9106 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  =  y )
28 id 20 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  { -u 1 ,  1 } )
2927, 28eqeltrd 2510 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  e.  { -u 1 ,  1 } )
30 oveq1 6088 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
3130eleq1d 2502 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( 1  x.  y
)  e.  { -u
1 ,  1 } ) )
3229, 31syl5ibr 213 . . . . . 6  |-  ( x  =  1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3326, 32jaoi 369 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( y  e.  { -u 1 ,  1 }  ->  (
x  x.  y )  e.  { -u 1 ,  1 } ) )
349, 33syl 16 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3534imp 419 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  y  e.  { -u 1 ,  1 } )  ->  ( x  x.  y )  e.  { -u 1 ,  1 } )
36 oveq2 6089 . . . . . . 7  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
37 divneg2 9738 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
383, 3, 4, 37mp3an 1279 . . . . . . . . 9  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
393div1i 9742 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4039negeqi 9299 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
4138, 40eqtr3i 2458 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
4241, 2eqeltri 2506 . . . . . . 7  |-  ( 1  /  -u 1 )  e. 
{ -u 1 ,  1 }
4336, 42syl6eqel 2524 . . . . . 6  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
44 oveq2 6089 . . . . . . 7  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
4539, 16eqeltri 2506 . . . . . . 7  |-  ( 1  /  1 )  e. 
{ -u 1 ,  1 }
4644, 45syl6eqel 2524 . . . . . 6  |-  ( x  =  1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
4743, 46jaoi 369 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
489, 47syl 16 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( 1  /  x )  e.  { -u 1 ,  1 } )
4948adantr 452 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  x  =/=  0
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
508, 35, 16, 49expcl2lem 11393 . 2  |-  ( (
-u 1  e.  { -u 1 ,  1 }  /\  -u 1  =/=  0  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
512, 5, 50mp3an12 1269 1  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3320   {cpr 3815  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995   -ucneg 9292    / cdiv 9677   ZZcz 10282   ^cexp 11382
This theorem is referenced by:  m1expcl  11404  lgseisenlem2  21134  m1expevenALT  24905  m1expaddsub  27398  psgnghm  27414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-exp 11383
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