Theorem m1exp1 12693

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 9683	. . . . . . 7 |- 2 e. ZZ
2		divides 12581	. . . . . . 7 |- ( ( 2 e. ZZ /\ N e. ZZ ) -> ( 2 || N <-> E. n e. ZZ ( n x. 2 ) = N ) )
3	1, 2	mpan 428	. . . . . 6 |- ( N e. ZZ -> ( 2 || N <-> E. n e. ZZ ( n x. 2 ) = N ) )
4		oveq2 6096	. . . . . . . . 9 |- ( N = ( n x. 2 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
5	4	eqcoms 2241	. . . . . . . 8 |- ( ( n x. 2 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( n x. 2 ) ) )
6		zcn 9660	. . . . . . . . . . 11 |- ( n e. ZZ -> n e. CC )
7		2cnd 9386	. . . . . . . . . . 11 |- ( n e. ZZ -> 2 e. CC )
8	6, 7	mulcomd 8353	. . . . . . . . . 10 |- ( n e. ZZ -> ( n x. 2 ) = ( 2 x. n ) )
9	8	oveq2d 6104	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = ( -u 1 ^ ( 2 x. n ) ) )
10		m1expeven 11044	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( 2 x. n ) ) = 1 )
11	9, 10	eqtrd 2271	. . . . . . . 8 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = 1 )
12	5, 11	sylan9eqr 2293	. . . . . . 7 |- ( ( n e. ZZ /\ ( n x. 2 ) = N ) -> ( -u 1 ^ N ) = 1 )
13	12	rexlimiva 2663	. . . . . 6 |- ( E. n e. ZZ ( n x. 2 ) = N -> ( -u 1 ^ N ) = 1 )
14	3, 13	biimtrdi 163	. . . . 5 |- ( N e. ZZ -> ( 2 || N -> ( -u 1 ^ N ) = 1 ) )
15	14	impcom 125	. . . 4 |- ( ( 2 || N /\ N e. ZZ ) -> ( -u 1 ^ N ) = 1 )
16		simpl 109	. . . 4 |- ( ( 2 || N /\ N e. ZZ ) -> 2 || N )
17	15, 16	2thd 175	. . 3 |- ( ( 2 || N /\ N e. ZZ ) -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )
18	17	expcom 116	. 2 |- ( N e. ZZ -> ( 2 || N -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) ) )
19		1ne0 9381	. . . . . 6 |- 1 =/= 0
20		eqcom 2240	. . . . . . 7 |- ( -u 1 = 1 <-> 1 = -u 1 )
21		ax-1cn 8278	. . . . . . . 8 |- 1 e. CC
22	21	eqnegi 9079	. . . . . . 7 |- ( 1 = -u 1 <-> 1 = 0 )
23	20, 22	bitri 184	. . . . . 6 |- ( -u 1 = 1 <-> 1 = 0 )
24	19, 23	nemtbir 2509	. . . . 5 |- -. -u 1 = 1
25		odd2np1 12665	. . . . . . . 8 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
26		oveq2 6096	. . . . . . . . . . 11 |- ( N = ( ( 2 x. n ) + 1 ) -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
27	26	eqcoms 2241	. . . . . . . . . 10 |- ( ( ( 2 x. n ) + 1 ) = N -> ( -u 1 ^ N ) = ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) )
28		neg1cn 9418	. . . . . . . . . . . . 13 |- -u 1 e. CC
29	28	a1i 9	. . . . . . . . . . . 12 |- ( n e. ZZ -> -u 1 e. CC )
30		neg1ap0 9422	. . . . . . . . . . . . 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 9785	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( 2 x. n ) e. ZZ )
35	29, 31, 34	expp1zapd 11141	. . . . . . . . . . 11 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) )
36	10	oveq1d 6103	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = ( 1 x. -u 1 ) )
37	28	mullidi 8335	. . . . . . . . . . . 12 |- ( 1 x. -u 1 ) = -u 1
38	36, 37	eqtrdi 2287	. . . . . . . . . . 11 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = -u 1 )
39	35, 38	eqtrd 2271	. . . . . . . . . 10 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = -u 1 )
40	27, 39	sylan9eqr 2293	. . . . . . . . 9 |- ( ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) -> ( -u 1 ^ N ) = -u 1 )
41	40	rexlimiva 2663	. . . . . . . 8 |- ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N -> ( -u 1 ^ N ) = -u 1 )
42	25, 41	biimtrdi 163	. . . . . . 7 |- ( N e. ZZ -> ( -. 2 || N -> ( -u 1 ^ N ) = -u 1 ) )
43	42	impcom 125	. . . . . 6 |- ( ( -. 2 || N /\ N e. ZZ ) -> ( -u 1 ^ N ) = -u 1 )
44	43	eqeq1d 2247	. . . . 5 |- ( ( -. 2 || N /\ N e. ZZ ) -> ( ( -u 1 ^ N ) = 1 <-> -u 1 = 1 ) )
45	24, 44	mtbiri 686	. . . 4 |- ( ( -. 2 || N /\ N e. ZZ ) -> -. ( -u 1 ^ N ) = 1 )
46		simpl 109	. . . 4 |- ( ( -. 2 || N /\ N e. ZZ ) -> -. 2 || N )
47	45, 46	2falsed 714	. . 3 |- ( ( -. 2 || N /\ N e. ZZ ) -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )
48	47	expcom 116	. 2 |- ( N e. ZZ -> ( -. 2 || N -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) ) )
49		zeo3 12660	. 2 |- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
50	18, 48, 49	mpjaod 730	1 |- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4131  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   -ucneg 8505   # cap 8917   2c2 9364   ZZcz 9655   ^cexp 10996   || cdvds 12579

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-exp 10997  df-dvds 12580

This theorem is used by:  2lgs  16428  2lgsoddprm  16437