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Theorem m1expeven 27725
Description: Exponentiation of negative one to an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Assertion
Ref Expression
m1expeven  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  1 )

Proof of Theorem m1expeven
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . 4  |-  ( x  =  0  ->  (
2  x.  x )  =  ( 2  x.  0 ) )
21oveq2d 5874 . . 3  |-  ( x  =  0  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  0 ) ) )
32eqeq1d 2291 . 2  |-  ( x  =  0  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  0 ) )  =  1 ) )
4 oveq2 5866 . . . 4  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
54oveq2d 5874 . . 3  |-  ( x  =  y  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  y ) ) )
65eqeq1d 2291 . 2  |-  ( x  =  y  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  y ) )  =  1 ) )
7 oveq2 5866 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
2  x.  x )  =  ( 2  x.  ( y  +  1 ) ) )
87oveq2d 5874 . . 3  |-  ( x  =  ( y  +  1 )  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) ) )
98eqeq1d 2291 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  1 ) )
10 oveq2 5866 . . . 4  |-  ( x  =  N  ->  (
2  x.  x )  =  ( 2  x.  N ) )
1110oveq2d 5874 . . 3  |-  ( x  =  N  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  N ) ) )
1211eqeq1d 2291 . 2  |-  ( x  =  N  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  N ) )  =  1 ) )
13 2cn 9816 . . . . 5  |-  2  e.  CC
1413mul01i 9002 . . . 4  |-  ( 2  x.  0 )  =  0
1514oveq2i 5869 . . 3  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  ( -u 1 ^ 0 )
16 neg1cn 9813 . . . 4  |-  -u 1  e.  CC
17 exp0 11108 . . . 4  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1816, 17ax-mp 8 . . 3  |-  ( -u
1 ^ 0 )  =  1
1915, 18eqtri 2303 . 2  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  1
2013a1i 10 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  2  e.  CC )
21 nn0cn 9975 . . . . . . . 8  |-  ( y  e.  NN0  ->  y  e.  CC )
2221adantr 451 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  y  e.  CC )
23 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
2423a1i 10 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  1  e.  CC )
2520, 22, 24adddid 8859 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  ( 2  x.  1 ) ) )
2620mulid1d 8852 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  1 )  =  2 )
2726oveq2d 5874 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( 2  x.  y )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2825, 27eqtrd 2315 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2928oveq2d 5874 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  ( -u 1 ^ ( ( 2  x.  y )  +  2 ) ) )
3024negcld 9144 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  -u 1  e.  CC )
31 2nn0 9982 . . . . . 6  |-  2  e.  NN0
3231a1i 10 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  2  e.  NN0 )
33 simpl 443 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  y  e.  NN0 )
3432, 33nn0mulcld 10023 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  y )  e.  NN0 )
3530, 32, 34expaddd 11247 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( ( 2  x.  y )  +  2 ) )  =  ( ( -u 1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) ) )
36 simpr 447 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  y ) )  =  1 )
3730sqvald 11242 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  ( -u 1  x.  -u 1 ) )
3824, 24mul2negd 9234 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1  x.  -u 1 )  =  ( 1  x.  1 ) )
3924mulid1d 8852 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 1  x.  1 )  =  1 )
4037, 38, 393eqtrd 2319 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  1 )
4136, 40oveq12d 5876 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  ( 1  x.  1 ) )
4241, 39eqtrd 2315 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  1 )
4329, 35, 423eqtrd 2319 . . 3  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  1 )
4443ex 423 . 2  |-  ( y  e.  NN0  ->  ( (
-u 1 ^ (
2  x.  y ) )  =  1  -> 
( -u 1 ^ (
2  x.  ( y  +  1 ) ) )  =  1 ) )
453, 6, 9, 12, 19, 44nn0ind 10108 1  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   2c2 9795   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  stirlinglem5  27827
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-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-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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