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Theorem m1expeven 27828
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 5882 . . . 4  |-  ( x  =  0  ->  (
2  x.  x )  =  ( 2  x.  0 ) )
21oveq2d 5890 . . 3  |-  ( x  =  0  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  0 ) ) )
32eqeq1d 2304 . 2  |-  ( x  =  0  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  0 ) )  =  1 ) )
4 oveq2 5882 . . . 4  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
54oveq2d 5890 . . 3  |-  ( x  =  y  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  y ) ) )
65eqeq1d 2304 . 2  |-  ( x  =  y  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  y ) )  =  1 ) )
7 oveq2 5882 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
2  x.  x )  =  ( 2  x.  ( y  +  1 ) ) )
87oveq2d 5890 . . 3  |-  ( x  =  ( y  +  1 )  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) ) )
98eqeq1d 2304 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  1 ) )
10 oveq2 5882 . . . 4  |-  ( x  =  N  ->  (
2  x.  x )  =  ( 2  x.  N ) )
1110oveq2d 5890 . . 3  |-  ( x  =  N  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  N ) ) )
1211eqeq1d 2304 . 2  |-  ( x  =  N  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  N ) )  =  1 ) )
13 2cn 9832 . . . . 5  |-  2  e.  CC
1413mul01i 9018 . . . 4  |-  ( 2  x.  0 )  =  0
1514oveq2i 5885 . . 3  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  ( -u 1 ^ 0 )
16 neg1cn 9829 . . . 4  |-  -u 1  e.  CC
17 exp0 11124 . . . 4  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1816, 17ax-mp 8 . . 3  |-  ( -u
1 ^ 0 )  =  1
1915, 18eqtri 2316 . 2  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  1
2013a1i 10 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  2  e.  CC )
21 nn0cn 9991 . . . . . . . 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 8811 . . . . . . . 8  |-  1  e.  CC
2423a1i 10 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  1  e.  CC )
2520, 22, 24adddid 8875 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  ( 2  x.  1 ) ) )
2620mulid1d 8868 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  1 )  =  2 )
2726oveq2d 5890 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( 2  x.  y )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2825, 27eqtrd 2328 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2928oveq2d 5890 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  ( -u 1 ^ ( ( 2  x.  y )  +  2 ) ) )
3024negcld 9160 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  -u 1  e.  CC )
31 2nn0 9998 . . . . . 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 10039 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  y )  e.  NN0 )
3530, 32, 34expaddd 11263 . . . 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 11258 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  ( -u 1  x.  -u 1 ) )
3824, 24mul2negd 9250 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1  x.  -u 1 )  =  ( 1  x.  1 ) )
3924mulid1d 8868 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 1  x.  1 )  =  1 )
4037, 38, 393eqtrd 2332 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  1 )
4136, 40oveq12d 5892 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  ( 1  x.  1 ) )
4241, 39eqtrd 2328 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  1 )
4329, 35, 423eqtrd 2332 . . 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 10124 1  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   -ucneg 9054   2c2 9811   NN0cn0 9981   ^cexp 11120
This theorem is referenced by:  stirlinglem5  27930
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-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-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121
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