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Theorem m1expeven 31786
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 6222 . . . 4  |-  ( x  =  0  ->  (
2  x.  x )  =  ( 2  x.  0 ) )
21oveq2d 6230 . . 3  |-  ( x  =  0  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  0 ) ) )
32eqeq1d 2394 . 2  |-  ( x  =  0  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  0 ) )  =  1 ) )
4 oveq2 6222 . . . 4  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
54oveq2d 6230 . . 3  |-  ( x  =  y  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  y ) ) )
65eqeq1d 2394 . 2  |-  ( x  =  y  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  y ) )  =  1 ) )
7 oveq2 6222 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
2  x.  x )  =  ( 2  x.  ( y  +  1 ) ) )
87oveq2d 6230 . . 3  |-  ( x  =  ( y  +  1 )  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) ) )
98eqeq1d 2394 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  1 ) )
10 oveq2 6222 . . . 4  |-  ( x  =  N  ->  (
2  x.  x )  =  ( 2  x.  N ) )
1110oveq2d 6230 . . 3  |-  ( x  =  N  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  N ) ) )
1211eqeq1d 2394 . 2  |-  ( x  =  N  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  N ) )  =  1 ) )
13 2t0e0 10626 . . . 4  |-  ( 2  x.  0 )  =  0
1413oveq2i 6225 . . 3  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  ( -u 1 ^ 0 )
15 neg1cn 10574 . . . 4  |-  -u 1  e.  CC
16 exp0 12092 . . . 4  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1715, 16ax-mp 5 . . 3  |-  ( -u
1 ^ 0 )  =  1
1814, 17eqtri 2421 . 2  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  1
19 2cnd 10543 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  2  e.  CC )
20 nn0cn 10740 . . . . . . . 8  |-  ( y  e.  NN0  ->  y  e.  CC )
2120adantr 463 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  y  e.  CC )
22 1cnd 9541 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  1  e.  CC )
2319, 21, 22adddid 9549 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  ( 2  x.  1 ) ) )
2419mulid1d 9542 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  1 )  =  2 )
2524oveq2d 6230 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( 2  x.  y )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2623, 25eqtrd 2433 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2726oveq2d 6230 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  ( -u 1 ^ ( ( 2  x.  y )  +  2 ) ) )
2822negcld 9849 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  -u 1  e.  CC )
29 2nn0 10747 . . . . . 6  |-  2  e.  NN0
3029a1i 11 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  2  e.  NN0 )
31 simpl 455 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  y  e.  NN0 )
3230, 31nn0mulcld 10792 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  y )  e.  NN0 )
3328, 30, 32expaddd 12233 . . . 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 ) ) )
34 simpr 459 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  y ) )  =  1 )
3528sqvald 12228 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  ( -u 1  x.  -u 1 ) )
3622, 22mul2negd 9947 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1  x.  -u 1 )  =  ( 1  x.  1 ) )
3722mulid1d 9542 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 1  x.  1 )  =  1 )
3835, 36, 373eqtrd 2437 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  1 )
3934, 38oveq12d 6232 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  ( 1  x.  1 ) )
4039, 37eqtrd 2433 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  1 )
4127, 33, 403eqtrd 2437 . . 3  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  1 )
4241ex 432 . 2  |-  ( y  e.  NN0  ->  ( (
-u 1 ^ (
2  x.  y ) )  =  1  -> 
( -u 1 ^ (
2  x.  ( y  +  1 ) ) )  =  1 ) )
433, 6, 9, 12, 18, 42nn0ind 10892 1  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836  (class class class)co 6214   CCcc 9419   0cc0 9421   1c1 9422    + caddc 9424    x. cmul 9426   -ucneg 9737   2c2 10520   NN0cn0 10730   ^cexp 12088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-2nd 6718  df-recs 6978  df-rdg 7012  df-er 7247  df-en 7454  df-dom 7455  df-sdom 7456  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-nn 10471  df-2 10529  df-n0 10731  df-z 10800  df-uz 11020  df-seq 12030  df-exp 12089
This theorem is referenced by:  stirlinglem5  32061
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