Theorem m1exp1 12646

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 9651	. . . . . . 7 |- 2 e. ZZ
2		divides 12534	. . . . . . 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 6083	. . . . . . . . 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 9628	. . . . . . . . . . 11 |- ( n e. ZZ -> n e. CC )
7		2cnd 9356	. . . . . . . . . . 11 |- ( n e. ZZ -> 2 e. CC )
8	6, 7	mulcomd 8337	. . . . . . . . . 10 |- ( n e. ZZ -> ( n x. 2 ) = ( 2 x. n ) )
9	8	oveq2d 6091	. . . . . . . . 9 |- ( n e. ZZ -> ( -u 1 ^ ( n x. 2 ) ) = ( -u 1 ^ ( 2 x. n ) ) )
10		m1expeven 11001	. . . . . . . . 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 9351	. . . . . 6 |- 1 =/= 0
20		eqcom 2240	. . . . . . 7 |- ( -u 1 = 1 <-> 1 = -u 1 )
21		ax-1cn 8262	. . . . . . . 8 |- 1 e. CC
22	21	eqnegi 9061	. . . . . . 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 12618	. . . . . . . 8 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
26		oveq2 6083	. . . . . . . . . . 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 9388	. . . . . . . . . . . . 13 |- -u 1 e. CC
29	28	a1i 9	. . . . . . . . . . . 12 |- ( n e. ZZ -> -u 1 e. CC )
30		neg1ap0 9392	. . . . . . . . . . . . 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 9753	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( 2 x. n ) e. ZZ )
35	29, 31, 34	expp1zapd 11098	. . . . . . . . . . 11 |- ( n e. ZZ -> ( -u 1 ^ ( ( 2 x. n ) + 1 ) ) = ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) )
36	10	oveq1d 6090	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( ( -u 1 ^ ( 2 x. n ) ) x. -u 1 ) = ( 1 x. -u 1 ) )
37	28	mullidi 8319	. . . . . . . . . . . 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 12613	. 2 |- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
50	18, 48, 49	mpjaod 730	1 |- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   -ucneg 8488   # cap 8899   2c2 9334   ZZcz 9623   ^cexp 10953   || cdvds 12532

This theorem was proved from 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-exp 10954  df-dvds 12533

This theorem is referenced by:  2lgs  16137  2lgsoddprm  16146