MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  m1expcl2 Unicode version

Theorem m1expcl2 11141
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 9066 . . 3  |-  -u 1  e.  _V
21prid1 3747 . 2  |-  -u 1  e.  { -u 1 ,  1 }
3 ax-1cn 8811 . . 3  |-  1  e.  CC
4 ax-1ne0 8822 . . 3  |-  1  =/=  0
53, 4negne0i 9137 . 2  |-  -u 1  =/=  0
6 neg1cn 9829 . . . 4  |-  -u 1  e.  CC
7 prssi 3787 . . . 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 3673 . . . . 5  |-  ( x  e.  { -u 1 ,  1 }  ->  ( x  =  -u 1  \/  x  =  1
) )
108sseli 3189 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  y  e.  CC )
1110mulm1d 9247 . . . . . . . 8  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  =  -u y
)
12 elpri 3673 . . . . . . . . 9  |-  ( y  e.  { -u 1 ,  1 }  ->  ( y  =  -u 1  \/  y  =  1
) )
13 negeq 9060 . . . . . . . . . . 11  |-  ( y  =  -u 1  ->  -u y  =  -u -u 1 )
143negnegi 9132 . . . . . . . . . . . 12  |-  -u -u 1  =  1
15 1ex 8849 . . . . . . . . . . . . 13  |-  1  e.  _V
1615prid2 3748 . . . . . . . . . . . 12  |-  1  e.  { -u 1 ,  1 }
1714, 16eqeltri 2366 . . . . . . . . . . 11  |-  -u -u 1  e.  { -u 1 ,  1 }
1813, 17syl6eqel 2384 . . . . . . . . . 10  |-  ( y  =  -u 1  ->  -u y  e.  { -u 1 ,  1 } )
19 negeq 9060 . . . . . . . . . . 11  |-  ( y  =  1  ->  -u y  =  -u 1 )
2019, 2syl6eqel 2384 . . . . . . . . . 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 2370 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  (
-u 1  x.  y
)  e.  { -u
1 ,  1 } )
24 oveq1 5881 . . . . . . . 8  |-  ( x  =  -u 1  ->  (
x  x.  y )  =  ( -u 1  x.  y ) )
2524eleq1d 2362 . . . . . . 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 8869 . . . . . . . 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 2370 . . . . . . 7  |-  ( y  e.  { -u 1 ,  1 }  ->  ( 1  x.  y )  e.  { -u 1 ,  1 } )
30 oveq1 5881 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
3130eleq1d 2362 . . . . . . 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 5882 . . . . . . 7  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
37 divneg2 9500 . . . . . . . . . 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 9504 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4039negeqi 9061 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
4138, 40eqtr3i 2318 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
4241, 2eqeltri 2366 . . . . . . 7  |-  ( 1  /  -u 1 )  e. 
{ -u 1 ,  1 }
4336, 42syl6eqel 2384 . . . . . 6  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { -u 1 ,  1 } )
44 oveq2 5882 . . . . . . 7  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
4539, 16eqeltri 2366 . . . . . . 7  |-  ( 1  /  1 )  e. 
{ -u 1 ,  1 }
4644, 45syl6eqel 2384 . . . . . 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 11131 . 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 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   {cpr 3654  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758   -ucneg 9054    / cdiv 9439   ZZcz 10040   ^cexp 11120
This theorem is referenced by:  m1expcl  11142  lgseisenlem2  20605  m1expaddsub  27524  psgnghm  27540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121
  Copyright terms: Public domain W3C validator