Theorem m1exp1 11634

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 9106	. . . . . . 7 |- 2 e. ZZ
2		divides 11531	. . . . . . 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 5790	. . . . . . . . 9 |- ( N = ( n x. 2 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
5	4	eqcoms 2143	. . . . . . . 8 |- ( ( n x. 2 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
6		zcn 9083	. . . . . . . . . . 11 |- ( n e. ZZ -> n e. CC )
7		2cnd 8817	. . . . . . . . . . 11 |- ( n e. ZZ -> 2 e. CC )
8	6, 7	mulcomd 7811	. . . . . . . . . 10 |- ( n e. ZZ -> ( n x. 2 ) = ( 2 x. n ) )
9	8	oveq2d 5798	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = ( -u 1 ^ ( 2 x. n ) ) )
10		m1expeven 10371	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( 2 x. n ) ) = 1 )
11	9, 10	eqtrd 2173	. . . . . . . 8 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = 1 )
12	5, 11	sylan9eqr 2195	. . . . . . 7 |- ( ( n e. ZZ /\ ( n x. 2 ) = N ) -> ( -u 1 ^ N ) = 1 )
13	12	rexlimiva 2547	. . . . . 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 8812	. . . . . 6 |- 1 =/= 0
20		eqcom 2142	. . . . . . 7 |- ( -u 1 = 1 <-> 1 = -u 1 )
21		ax-1cn 7737	. . . . . . . 8 |- 1 e. CC
22	21	eqnegi 8525	. . . . . . 7 |- ( 1 = -u 1 <-> 1 = 0 )
23	20, 22	bitri 183	. . . . . 6 |- ( -u 1 = 1 <-> 1 = 0 )
24	19, 23	nemtbir 2398	. . . . 5 |- -. -u 1 = 1
25		odd2np1 11606	. . . . . . . 8 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
26		oveq2 5790	. . . . . . . . . . 11 |- ( N = ( ( 2 x. n ) + 1 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
27	26	eqcoms 2143	. . . . . . . . . 10 |- ( ( ( 2 x. n ) + 1 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
28		neg1cn 8849	. . . . . . . . . . . . 13 |- -u 1 e. CC
29	28	a1i 9	. . . . . . . . . . . 12 |- ( n e. ZZ -> -u 1 e. CC )
30		neg1ap0 8853	. . . . . . . . . . . . 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 9203	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( 2 x. n ) e. ZZ )
35	29, 31, 34	expp1zapd 10464	. . . . . . . . . . 11 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) )
36	10	oveq1d 5797	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = ( 1 x. -u 1 ) )
37	28	mulid2i 7793	. . . . . . . . . . . 12 |- ( 1 x. -u 1 ) = -u 1
38	36, 37	eqtrdi 2189	. . . . . . . . . . 11 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = -u 1 )
39	35, 38	eqtrd 2173	. . . . . . . . . 10 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = -u 1 )
40	27, 39	sylan9eqr 2195	. . . . . . . . 9 |- ( ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) -> ( -u 1 ^ N ) = -u 1 )
41	40	rexlimiva 2547	. . . . . . . 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 2149	. . . . 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 11601	. 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 1332   e. wcel 1481   E.wrex 2418   class class class wbr 3937  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   -ucneg 7958   # cap 8367   2c2 8795   ZZcz 9078   ^cexp 10323   || cdvds 11529

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-xor 1355  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250  df-exp 10324  df-dvds 11530

This theorem is referenced by: (None)