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Theorem m1expcl2 12332
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 9893 . . 3  |-  -u 1  e.  _V
21prid1 4071 . 2  |-  -u 1  e.  { -u 1 ,  1 }
3 neg1ne0 10737 . 2  |-  -u 1  =/=  0
4 neg1cn 10735 . . . 4  |-  -u 1  e.  CC
5 ax-1cn 9615 . . . 4  |-  1  e.  CC
6 prssi 4119 . . . 4  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
74, 5, 6mp2an 686 . . 3  |-  { -u
1 ,  1 } 
C_  CC
8 elpri 3976 . . . . 5  |-  ( x  e.  { -u 1 ,  1 }  ->  ( x  =  -u 1  \/  x  =  1
) )
97sseli 3414 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  CC )
109mulm1d 10091 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  =  -u y
)
11 elpri 3976 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  ( y  =  -u 1  \/  y  =  1
) )
12 negeq 9887 . . . . . . . . . . 11  |-  ( y  =  -u 1  ->  -u y  =  -u -u 1 )
13 negneg1e1 10739 . . . . . . . . . . . 12  |-  -u -u 1  =  1
14 1ex 9656 . . . . . . . . . . . . 13  |-  1  e.  _V
1514prid2 4072 . . . . . . . . . . . 12  |-  1  e.  { -u 1 ,  1 }
1613, 15eqeltri 2545 . . . . . . . . . . 11  |-  -u -u 1  e.  { -u 1 ,  1 }
1712, 16syl6eqel 2557 . . . . . . . . . 10  |-  ( y  =  -u 1  ->  -u y  e.  { -u 1 ,  1 } )
18 negeq 9887 . . . . . . . . . . 11  |-  ( y  =  1  ->  -u y  =  -u 1 )
1918, 2syl6eqel 2557 . . . . . . . . . 10  |-  ( y  =  1  ->  -u y  e.  { -u 1 ,  1 } )
2017, 19jaoi 386 . . . . . . . . 9  |-  ( ( y  =  -u 1  \/  y  =  1
)  ->  -u y  e. 
{ -u 1 ,  1 } )
2111, 20syl 17 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  -u y  e.  { -u 1 ,  1 } )
2210, 21eqeltrd 2549 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  e.  { -u
1 ,  1 } )
23 oveq1 6315 . . . . . . . 8  |-  ( x  =  -u 1  ->  (
x  x.  y )  =  ( -u 1  x.  y ) )
2423eleq1d 2533 . . . . . . 7  |-  ( x  =  -u 1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( -u 1  x.  y
)  e.  { -u
1 ,  1 } ) )
2522, 24syl5ibr 229 . . . . . 6  |-  ( x  =  -u 1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
269mulid2d 9679 . . . . . . . 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 2549 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  e.  { -u 1 ,  1 } )
29 oveq1 6315 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
3029eleq1d 2533 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  e.  { -u
1 ,  1 }  <-> 
( 1  x.  y
)  e.  { -u
1 ,  1 } ) )
3128, 30syl5ibr 229 . . . . . 6  |-  ( x  =  1  ->  (
y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3225, 31jaoi 386 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( y  e.  { -u 1 ,  1 }  ->  (
x  x.  y )  e.  { -u 1 ,  1 } ) )
338, 32syl 17 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( y  e.  { -u
1 ,  1 }  ->  ( x  x.  y )  e.  { -u 1 ,  1 } ) )
3433imp 436 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  y  e.  { -u 1 ,  1 } )  ->  ( x  x.  y )  e.  { -u 1 ,  1 } )
35 oveq2 6316 . . . . . . 7  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
36 ax-1ne0 9626 . . . . . . . . . 10  |-  1  =/=  0
37 divneg2 10353 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
385, 5, 36, 37mp3an 1390 . . . . . . . . 9  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
39 1div1e1 10322 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4039negeqi 9888 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
4138, 40eqtr3i 2495 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
4241, 2eqeltri 2545 . . . . . . 7  |-  ( 1  /  -u 1 )  e. 
{ -u 1 ,  1 }
4335, 42syl6eqel 2557 . . . . . 6  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
44 oveq2 6316 . . . . . . 7  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
4539, 15eqeltri 2545 . . . . . . 7  |-  ( 1  /  1 )  e. 
{ -u 1 ,  1 }
4644, 45syl6eqel 2557 . . . . . 6  |-  ( x  =  1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
4743, 46jaoi 386 . . . . 5  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
488, 47syl 17 . . . 4  |-  ( x  e.  { -u 1 ,  1 }  ->  ( 1  /  x )  e.  { -u 1 ,  1 } )
4948adantr 472 . . 3  |-  ( ( x  e.  { -u
1 ,  1 }  /\  x  =/=  0
)  ->  ( 1  /  x )  e. 
{ -u 1 ,  1 } )
507, 34, 15, 49expcl2lem 12322 . 2  |-  ( (
-u 1  e.  { -u 1 ,  1 }  /\  -u 1  =/=  0  /\  N  e.  ZZ )  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
512, 3, 50mp3an12 1380 1  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    = wceq 1452    e. wcel 1904    =/= wne 2641    C_ wss 3390   {cpr 3961  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    x. cmul 9562   -ucneg 9881    / cdiv 10291   ZZcz 10961   ^cexp 12310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-seq 12252  df-exp 12311
This theorem is referenced by:  m1expcl  12333  m1expeven  12357  m1expaddsub  17217  psgnran  17234  psgnghm  19225  lgseisenlem2  24357  madjusmdetlem4  28730
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