Theorem m1expeven 9523

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 8356	. . . 4 |- ( N e. ZZ -> N e. CC )
2	1	2timesd 8273	. . 3 |- ( N e. ZZ -> ( 2 x. N ) = ( N + N ) )
3	2	oveq2d 5548	. 2 |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = ( -u 1 ^ ( N + N ) ) )
4		neg1cn 8144	. . . 4 |- -u 1 e. CC
5		neg1ap0 8148	. . . 4 |- -u 1 # 0
6		expaddzap 9520	. . . 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 426	. . 3 |- ( ( N e. ZZ /\ N e. ZZ ) -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
8	7	anidms 389	. 2 |- ( N e. ZZ -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
9		m1expcl2 9498	. . 3 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
10		neg1rr 8145	. . . . . 6 |- -u 1 e. RR
11		reexpclzap 9496	. . . . . 6 |- ( ( -u 1 e. RR /\ -u 1 # 0 /\ N e. ZZ ) -> ( -u 1 ^ N ) e. RR )
12	10, 5, 11	mp3an12 1258	. . . . 5 |- ( N e. ZZ -> ( -u 1 ^ N ) e. RR )
13		elprg 3418	. . . . 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 5541	. . . . . . 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 389	. . . . . 6 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( -u 1 x. -u 1 ) )
17		neg1mulneg1e1 8243	. . . . . 6 |- ( -u 1 x. -u 1 ) = 1
18	16, 17	syl6eq 2129	. . . . 5 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
19		oveq12 5541	. . . . . . 7 |- ( ( ( -u 1 ^ N ) = 1 /\ ( -u 1 ^ N ) = 1 ) -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( 1 x. 1 ) )
20	19	anidms 389	. . . . . 6 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( 1 x. 1 ) )
21		1t1e1 8184	. . . . . 6 |- ( 1 x. 1 ) = 1
22	20, 21	syl6eq 2129	. . . . 5 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
23	18, 22	jaoi 668	. . . 4 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
24	14, 23	syl6bi 161	. . 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 2117	1 |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = 1 )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   {cpr 3399   class class class wbr 3785  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   -ucneg 7280   # cap 7681   2c2 8089   ZZcz 8351   ^cexp 9475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476

This theorem is referenced by:  m1expe  10299  m1expo  10300  m1exp1  10301