Theorem m1exp1 11860

Description: Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)

Assertion

Ref	Expression
m1exp1	|- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )

Proof of Theorem m1exp1

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2z 9240	. . . . . . 7 |- 2 e. ZZ
2		divides 11751	. . . . . . 7 |- ( ( 2 e. ZZ /\ N e. ZZ ) -> ( 2 || N <-> E. n e. ZZ ( n x. 2 ) = N ) )
3	1, 2	mpan 422	. . . . . 6 |- ( N e. ZZ -> ( 2 || N <-> E. n e. ZZ ( n x. 2 ) = N ) )
4		oveq2 5861	. . . . . . . . 9 |- ( N = ( n x. 2 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
5	4	eqcoms 2173	. . . . . . . 8 |- ( ( n x. 2 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
6		zcn 9217	. . . . . . . . . . 11 |- ( n e. ZZ -> n e. CC )
7		2cnd 8951	. . . . . . . . . . 11 |- ( n e. ZZ -> 2 e. CC )
8	6, 7	mulcomd 7941	. . . . . . . . . 10 |- ( n e. ZZ -> ( n x. 2 ) = ( 2 x. n ) )
9	8	oveq2d 5869	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = ( -u 1 ^ ( 2 x. n ) ) )
10		m1expeven 10523	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( 2 x. n ) ) = 1 )
11	9, 10	eqtrd 2203	. . . . . . . 8 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = 1 )
12	5, 11	sylan9eqr 2225	. . . . . . 7 |- ( ( n e. ZZ /\ ( n x. 2 ) = N ) -> ( -u 1 ^ N ) = 1 )
13	12	rexlimiva 2582	. . . . . 6 |- ( E. n e. ZZ ( n x. 2 ) = N -> ( -u 1 ^ N ) = 1 )
14	3, 13	syl6bi 162	. . . . 5 |- ( N e. ZZ -> ( 2 || N -> ( -u 1 ^ N ) = 1 ) )
15	14	impcom 124	. . . 4 |- ( ( 2 || N /\ N e. ZZ ) -> ( -u 1 ^ N ) = 1 )
16		simpl 108	. . . 4 |- ( ( 2 || N /\ N e. ZZ ) -> 2 || N )
17	15, 16	2thd 174	. . 3 |- ( ( 2 || N /\ N e. ZZ ) -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )
18	17	expcom 115	. 2 |- ( N e. ZZ -> ( 2 || N -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) ) )
19		1ne0 8946	. . . . . 6 |- 1 =/= 0
20		eqcom 2172	. . . . . . 7 |- ( -u 1 = 1 <-> 1 = -u 1 )
21		ax-1cn 7867	. . . . . . . 8 |- 1 e. CC
22	21	eqnegi 8658	. . . . . . 7 |- ( 1 = -u 1 <-> 1 = 0 )
23	20, 22	bitri 183	. . . . . 6 |- ( -u 1 = 1 <-> 1 = 0 )
24	19, 23	nemtbir 2429	. . . . 5 |- -. -u 1 = 1
25		odd2np1 11832	. . . . . . . 8 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
26		oveq2 5861	. . . . . . . . . . 11 |- ( N = ( ( 2 x. n ) + 1 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
27	26	eqcoms 2173	. . . . . . . . . 10 |- ( ( ( 2 x. n ) + 1 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
28		neg1cn 8983	. . . . . . . . . . . . 13 |- -u 1 e. CC
29	28	a1i 9	. . . . . . . . . . . 12 |- ( n e. ZZ -> -u 1 e. CC )
30		neg1ap0 8987	. . . . . . . . . . . . 13 |- -u 1 # 0
31	30	a1i 9	. . . . . . . . . . . 12 |- ( n e. ZZ -> -u 1 # 0 )
32	1	a1i 9	. . . . . . . . . . . . 13 |- ( n e. ZZ -> 2 e. ZZ )
33		id 19	. . . . . . . . . . . . 13 |- ( n e. ZZ -> n e. ZZ )
34	32, 33	zmulcld 9340	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( 2 x. n ) e. ZZ )
35	29, 31, 34	expp1zapd 10618	. . . . . . . . . . 11 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) )
36	10	oveq1d 5868	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = ( 1 x. -u 1 ) )
37	28	mulid2i 7923	. . . . . . . . . . . 12 |- ( 1 x. -u 1 ) = -u 1
38	36, 37	eqtrdi 2219	. . . . . . . . . . 11 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = -u 1 )
39	35, 38	eqtrd 2203	. . . . . . . . . 10 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = -u 1 )
40	27, 39	sylan9eqr 2225	. . . . . . . . 9 |- ( ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) -> ( -u 1 ^ N ) = -u 1 )
41	40	rexlimiva 2582	. . . . . . . 8 |- ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N -> ( -u 1 ^ N ) = -u 1 )
42	25, 41	syl6bi 162	. . . . . . 7 |- ( N e. ZZ -> ( -. 2 || N -> ( -u 1 ^ N ) = -u 1 ) )
43	42	impcom 124	. . . . . 6 |- ( ( -. 2 || N /\ N e. ZZ ) -> ( -u 1 ^ N ) = -u 1 )
44	43	eqeq1d 2179	. . . . 5 |- ( ( -. 2 || N /\ N e. ZZ ) -> ( ( -u 1 ^ N ) = 1 <-> -u 1 = 1 ) )
45	24, 44	mtbiri 670	. . . 4 |- ( ( -. 2 || N /\ N e. ZZ ) -> -. ( -u 1 ^ N ) = 1 )
46		simpl 108	. . . 4 |- ( ( -. 2 || N /\ N e. ZZ ) -> -. 2 || N )
47	45, 46	2falsed 697	. . 3 |- ( ( -. 2 || N /\ N e. ZZ ) -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )
48	47	expcom 115	. 2 |- ( N e. ZZ -> ( -. 2 || N -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) ) )
49		zeo3 11827	. 2 |- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
50	18, 48, 49	mpjaod 713	1 |- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   -ucneg 8091   # cap 8500   2c2 8929   ZZcz 9212   ^cexp 10475   || cdvds 11749

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-xor 1371  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  df-dvds 11750

This theorem is referenced by: (None)