Theorem m1expeven 10181

Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)

Assertion

Ref	Expression
m1expeven	|- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = 1 )

Proof of Theorem m1expeven

Step	Hyp	Ref	Expression
1		zcn 8911	. . . 4 |- ( N e. ZZ -> N e. CC )
2	1	2timesd 8814	. . 3 |- ( N e. ZZ -> ( 2 x. N ) = ( N + N ) )
3	2	oveq2d 5722	. 2 |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = ( -u 1 ^ ( N + N ) ) )
4		neg1cn 8683	. . . 4 |- -u 1 e. CC
5		neg1ap0 8687	. . . 4 |- -u 1 # 0
6		expaddzap 10178	. . . 4 |- ( ( ( -u 1 e. CC /\ -u 1 # 0 ) /\ ( N e. ZZ /\ N e. ZZ ) ) -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
7	4, 5, 6	mpanl12 430	. . 3 |- ( ( N e. ZZ /\ N e. ZZ ) -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
8	7	anidms 392	. 2 |- ( N e. ZZ -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
9		m1expcl2 10156	. . 3 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
10		neg1rr 8684	. . . . . 6 |- -u 1 e. RR
11		reexpclzap 10154	. . . . . 6 |- ( ( -u 1 e. RR /\ -u 1 # 0 /\ N e. ZZ ) -> ( -u 1 ^ N ) e. RR )
12	10, 5, 11	mp3an12 1273	. . . . 5 |- ( N e. ZZ -> ( -u 1 ^ N ) e. RR )
13		elprg 3494	. . . . 5 |- ( ( -u 1 ^ N ) e. RR -> ( ( -u 1 ^ N ) e. { -u 1 , 1 } <-> ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) ) )
14	12, 13	syl 14	. . . 4 |- ( N e. ZZ -> ( ( -u 1 ^ N ) e. { -u 1 , 1 } <-> ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) ) )
15		oveq12 5715	. . . . . . 7 |- ( ( ( -u 1 ^ N ) = -u 1 /\ ( -u 1 ^ N ) = -u 1 ) -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( -u 1 x. -u 1 ) )
16	15	anidms 392	. . . . . 6 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( -u 1 x. -u 1 ) )
17		neg1mulneg1e1 8784	. . . . . 6 |- ( -u 1 x. -u 1 ) = 1
18	16, 17	syl6eq 2148	. . . . 5 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
19		oveq12 5715	. . . . . . 7 |- ( ( ( -u 1 ^ N ) = 1 /\ ( -u 1 ^ N ) = 1 ) -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( 1 x. 1 ) )
20	19	anidms 392	. . . . . 6 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( 1 x. 1 ) )
21		1t1e1 8724	. . . . . 6 |- ( 1 x. 1 ) = 1
22	20, 21	syl6eq 2148	. . . . 5 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
23	18, 22	jaoi 677	. . . 4 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
24	14, 23	syl6bi 162	. . 3 |- ( N e. ZZ -> ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 ) )
25	9, 24	mpd 13	. 2 |- ( N e. ZZ -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
26	3, 8, 25	3eqtrd 2136	1 |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = 1 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670   = wceq 1299   e. wcel 1448   {cpr 3475   class class class wbr 3875  (class class class)co 5706   CCcc 7498   RRcr 7499   0cc0 7500   1c1 7501   + caddc 7503   x. cmul 7505   -ucneg 7805   # cap 8209   2c2 8629   ZZcz 8906   ^cexp 10133

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-n0 8830  df-z 8907  df-uz 9177  df-seqfrec 10060  df-exp 10134

This theorem is referenced by:  m1expe  11391  m1expo  11392  m1exp1  11393