Theorem m1expeven 10523

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 9217	. . . 4 |- ( N e. ZZ -> N e. CC )
2	1	2timesd 9120	. . 3 |- ( N e. ZZ -> ( 2 x. N ) = ( N + N ) )
3	2	oveq2d 5869	. 2 |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = ( -u 1 ^ ( N + N ) ) )
4		neg1cn 8983	. . . 4 |- -u 1 e. CC
5		neg1ap0 8987	. . . 4 |- -u 1 # 0
6		expaddzap 10520	. . . 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 434	. . 3 |- ( ( N e. ZZ /\ N e. ZZ ) -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
8	7	anidms 395	. 2 |- ( N e. ZZ -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
9		m1expcl2 10498	. . 3 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
10		neg1rr 8984	. . . . . 6 |- -u 1 e. RR
11		reexpclzap 10496	. . . . . 6 |- ( ( -u 1 e. RR /\ -u 1 # 0 /\ N e. ZZ ) -> ( -u 1 ^ N ) e. RR )
12	10, 5, 11	mp3an12 1322	. . . . 5 |- ( N e. ZZ -> ( -u 1 ^ N ) e. RR )
13		elprg 3603	. . . . 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 5862	. . . . . . 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 395	. . . . . 6 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( -u 1 x. -u 1 ) )
17		neg1mulneg1e1 9090	. . . . . 6 |- ( -u 1 x. -u 1 ) = 1
18	16, 17	eqtrdi 2219	. . . . 5 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
19		oveq12 5862	. . . . . . 7 |- ( ( ( -u 1 ^ N ) = 1 /\ ( -u 1 ^ N ) = 1 ) -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( 1 x. 1 ) )
20	19	anidms 395	. . . . . 6 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = ( 1 x. 1 ) )
21		1t1e1 9030	. . . . . 6 |- ( 1 x. 1 ) = 1
22	20, 21	eqtrdi 2219	. . . . 5 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
23	18, 22	jaoi 711	. . . 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 2207	1 |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = 1 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   {cpr 3584   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   -ucneg 8091   # cap 8500   2c2 8929   ZZcz 9212   ^cexp 10475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476

This theorem is referenced by:  m1expe  11858  m1expo  11859  m1exp1  11860