Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  m1expeven Structured version   Unicode version

Theorem m1expeven 27715
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 6092 . . . 4  |-  ( x  =  0  ->  (
2  x.  x )  =  ( 2  x.  0 ) )
21oveq2d 6100 . . 3  |-  ( x  =  0  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  0 ) ) )
32eqeq1d 2446 . 2  |-  ( x  =  0  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  0 ) )  =  1 ) )
4 oveq2 6092 . . . 4  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
54oveq2d 6100 . . 3  |-  ( x  =  y  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  y ) ) )
65eqeq1d 2446 . 2  |-  ( x  =  y  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  y ) )  =  1 ) )
7 oveq2 6092 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
2  x.  x )  =  ( 2  x.  ( y  +  1 ) ) )
87oveq2d 6100 . . 3  |-  ( x  =  ( y  +  1 )  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) ) )
98eqeq1d 2446 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  1 ) )
10 oveq2 6092 . . . 4  |-  ( x  =  N  ->  (
2  x.  x )  =  ( 2  x.  N ) )
1110oveq2d 6100 . . 3  |-  ( x  =  N  ->  ( -u 1 ^ ( 2  x.  x ) )  =  ( -u 1 ^ ( 2  x.  N ) ) )
1211eqeq1d 2446 . 2  |-  ( x  =  N  ->  (
( -u 1 ^ (
2  x.  x ) )  =  1  <->  ( -u 1 ^ ( 2  x.  N ) )  =  1 ) )
13 2cn 10075 . . . . 5  |-  2  e.  CC
1413mul01i 9261 . . . 4  |-  ( 2  x.  0 )  =  0
1514oveq2i 6095 . . 3  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  ( -u 1 ^ 0 )
16 neg1cn 10072 . . . 4  |-  -u 1  e.  CC
17 exp0 11391 . . . 4  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1816, 17ax-mp 5 . . 3  |-  ( -u
1 ^ 0 )  =  1
1915, 18eqtri 2458 . 2  |-  ( -u
1 ^ ( 2  x.  0 ) )  =  1
2013a1i 11 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  2  e.  CC )
21 nn0cn 10236 . . . . . . . 8  |-  ( y  e.  NN0  ->  y  e.  CC )
2221adantr 453 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  y  e.  CC )
23 ax-1cn 9053 . . . . . . . 8  |-  1  e.  CC
2423a1i 11 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  1  e.  CC )
2520, 22, 24adddid 9117 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  ( 2  x.  1 ) ) )
2620mulid1d 9110 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  1 )  =  2 )
2726oveq2d 6100 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( 2  x.  y )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2825, 27eqtrd 2470 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  ( y  +  1 ) )  =  ( ( 2  x.  y
)  +  2 ) )
2928oveq2d 6100 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  ( -u 1 ^ ( ( 2  x.  y )  +  2 ) ) )
3024negcld 9403 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  -u 1  e.  CC )
31 2nn0 10243 . . . . . 6  |-  2  e.  NN0
3231a1i 11 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  2  e.  NN0 )
33 simpl 445 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  y  e.  NN0 )
3432, 33nn0mulcld 10284 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 2  x.  y )  e.  NN0 )
3530, 32, 34expaddd 11530 . . . 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 449 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  y ) )  =  1 )
3730sqvald 11525 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  ( -u 1  x.  -u 1 ) )
3824, 24mul2negd 9493 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1  x.  -u 1 )  =  ( 1  x.  1 ) )
3924mulid1d 9110 . . . . . . 7  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( 1  x.  1 )  =  1 )
4037, 38, 393eqtrd 2474 . . . . . 6  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ 2 )  =  1 )
4136, 40oveq12d 6102 . . . . 5  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  ( 1  x.  1 ) )
4241, 39eqtrd 2470 . . . 4  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( ( -u
1 ^ ( 2  x.  y ) )  x.  ( -u 1 ^ 2 ) )  =  1 )
4329, 35, 423eqtrd 2474 . . 3  |-  ( ( y  e.  NN0  /\  ( -u 1 ^ (
2  x.  y ) )  =  1 )  ->  ( -u 1 ^ ( 2  x.  ( y  +  1 ) ) )  =  1 )
4443ex 425 . 2  |-  ( y  e.  NN0  ->  ( (
-u 1 ^ (
2  x.  y ) )  =  1  -> 
( -u 1 ^ (
2  x.  ( y  +  1 ) ) )  =  1 ) )
453, 6, 9, 12, 19, 44nn0ind 10371 1  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000   -ucneg 9297   2c2 10054   NN0cn0 10226   ^cexp 11387
This theorem is referenced by:  stirlinglem5  27817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329  df-exp 11388
  Copyright terms: Public domain W3C validator