Theorem m1exp1 11838

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 9219	. . . . . . 7 |- 2 e. ZZ
2		divides 11729	. . . . . . 7 |- ( ( 2 e. ZZ /\ N e. ZZ ) -> ( 2 || N <-> E. n e. ZZ ( n x. 2 ) = N ) )
3	1, 2	mpan 421	. . . . . 6 |- ( N e. ZZ -> ( 2 || N <-> E. n e. ZZ ( n x. 2 ) = N ) )
4		oveq2 5850	. . . . . . . . 9 |- ( N = ( n x. 2 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
5	4	eqcoms 2168	. . . . . . . 8 |- ( ( n x. 2 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
6		zcn 9196	. . . . . . . . . . 11 |- ( n e. ZZ -> n e. CC )
7		2cnd 8930	. . . . . . . . . . 11 |- ( n e. ZZ -> 2 e. CC )
8	6, 7	mulcomd 7920	. . . . . . . . . 10 |- ( n e. ZZ -> ( n x. 2 ) = ( 2 x. n ) )
9	8	oveq2d 5858	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = ( -u 1 ^ ( 2 x. n ) ) )
10		m1expeven 10502	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( 2 x. n ) ) = 1 )
11	9, 10	eqtrd 2198	. . . . . . . 8 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = 1 )
12	5, 11	sylan9eqr 2221	. . . . . . 7 |- ( ( n e. ZZ /\ ( n x. 2 ) = N ) -> ( -u 1 ^ N ) = 1 )
13	12	rexlimiva 2578	. . . . . 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 8925	. . . . . 6 |- 1 =/= 0
20		eqcom 2167	. . . . . . 7 |- ( -u 1 = 1 <-> 1 = -u 1 )
21		ax-1cn 7846	. . . . . . . 8 |- 1 e. CC
22	21	eqnegi 8637	. . . . . . 7 |- ( 1 = -u 1 <-> 1 = 0 )
23	20, 22	bitri 183	. . . . . 6 |- ( -u 1 = 1 <-> 1 = 0 )
24	19, 23	nemtbir 2425	. . . . 5 |- -. -u 1 = 1
25		odd2np1 11810	. . . . . . . 8 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
26		oveq2 5850	. . . . . . . . . . 11 |- ( N = ( ( 2 x. n ) + 1 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
27	26	eqcoms 2168	. . . . . . . . . 10 |- ( ( ( 2 x. n ) + 1 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
28		neg1cn 8962	. . . . . . . . . . . . 13 |- -u 1 e. CC
29	28	a1i 9	. . . . . . . . . . . 12 |- ( n e. ZZ -> -u 1 e. CC )
30		neg1ap0 8966	. . . . . . . . . . . . 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 9319	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( 2 x. n ) e. ZZ )
35	29, 31, 34	expp1zapd 10597	. . . . . . . . . . 11 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) )
36	10	oveq1d 5857	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = ( 1 x. -u 1 ) )
37	28	mulid2i 7902	. . . . . . . . . . . 12 |- ( 1 x. -u 1 ) = -u 1
38	36, 37	eqtrdi 2215	. . . . . . . . . . 11 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = -u 1 )
39	35, 38	eqtrd 2198	. . . . . . . . . 10 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = -u 1 )
40	27, 39	sylan9eqr 2221	. . . . . . . . 9 |- ( ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) -> ( -u 1 ^ N ) = -u 1 )
41	40	rexlimiva 2578	. . . . . . . 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 2174	. . . . 5 |- ( ( -. 2 || N /\ N e. ZZ ) -> ( ( -u 1 ^ N ) = 1 <-> -u 1 = 1 ) )
45	24, 44	mtbiri 665	. . . 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 692	. . 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 11805	. 2 |- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
50	18, 48, 49	mpjaod 708	1 |- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   -ucneg 8070   # cap 8479   2c2 8908   ZZcz 9191   ^cexp 10454   || cdvds 11727

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381  df-exp 10455  df-dvds 11728

This theorem is referenced by: (None)