Theorem m1expcl2 12167

Description: Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)

Assertion

Ref	Expression
m1expcl2	|- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )

Proof of Theorem m1expcl2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		negex 9823	. . 3 |- -u 1 e. _V
2	1	prid1 4123	. 2 |- -u 1 e. { -u 1 , 1 }
3		neg1ne0 10647	. 2 |- -u 1 =/= 0
4		neg1cn 10645	. . . 4 |- -u 1 e. CC
5		ax-1cn 9553	. . . 4 |- 1 e. CC
6		prssi 4171	. . . 4 |- ( ( -u 1 e. CC /\ 1 e. CC ) -> { -u 1 , 1 } C_ CC )
7	4, 5, 6	mp2an 672	. . 3 |- { -u 1 , 1 } C_ CC
8		elpri 4034	. . . . 5 |- ( x e. { -u 1 , 1 } -> ( x = -u 1 \/ x = 1 ) )
9	7	sseli 3485	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> y e. CC )
10	9	mulm1d 10014	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) = -u y )
11		elpri 4034	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> ( y = -u 1 \/ y = 1 ) )
12		negeq 9817	. . . . . . . . . . 11 |- ( y = -u 1 -> -u y = -u -u 1 )
13		negneg1e1 10649	. . . . . . . . . . . 12 |- -u -u 1 = 1
14		1ex 9594	. . . . . . . . . . . . 13 |- 1 e. _V
15	14	prid2 4124	. . . . . . . . . . . 12 |- 1 e. { -u 1 , 1 }
16	13, 15	eqeltri 2527	. . . . . . . . . . 11 |- -u -u 1 e. { -u 1 , 1 }
17	12, 16	syl6eqel 2539	. . . . . . . . . 10 |- ( y = -u 1 -> -u y e. { -u 1 , 1 } )
18		negeq 9817	. . . . . . . . . . 11 |- ( y = 1 -> -u y = -u 1 )
19	18, 2	syl6eqel 2539	. . . . . . . . . 10 |- ( y = 1 -> -u y e. { -u 1 , 1 } )
20	17, 19	jaoi 379	. . . . . . . . 9 |- ( ( y = -u 1 \/ y = 1 ) -> -u y e. { -u 1 , 1 } )
21	11, 20	syl 16	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> -u y e. { -u 1 , 1 } )
22	10, 21	eqeltrd 2531	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) e. { -u 1 , 1 } )
23		oveq1 6288	. . . . . . . 8 |- ( x = -u 1 -> ( x x. y ) = ( -u 1 x. y ) )
24	23	eleq1d 2512	. . . . . . 7 |- ( x = -u 1 -> ( ( x x. y ) e. { -u 1 , 1 } <-> ( -u 1 x. y ) e. { -u 1 , 1 } ) )
25	22, 24	syl5ibr 221	. . . . . 6 |- ( x = -u 1 -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
26	9	mulid2d 9617	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> ( 1 x. y ) = y )
27		id 22	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> y e. { -u 1 , 1 } )
28	26, 27	eqeltrd 2531	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( 1 x. y ) e. { -u 1 , 1 } )
29		oveq1 6288	. . . . . . . 8 |- ( x = 1 -> ( x x. y ) = ( 1 x. y ) )
30	29	eleq1d 2512	. . . . . . 7 |- ( x = 1 -> ( ( x x. y ) e. { -u 1 , 1 } <-> ( 1 x. y ) e. { -u 1 , 1 } ) )
31	28, 30	syl5ibr 221	. . . . . 6 |- ( x = 1 -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
32	25, 31	jaoi 379	. . . . 5 |- ( ( x = -u 1 \/ x = 1 ) -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
33	8, 32	syl 16	. . . 4 |- ( x e. { -u 1 , 1 } -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
34	33	imp 429	. . 3 |- ( ( x e. { -u 1 , 1 } /\ y e. { -u 1 , 1 } ) -> ( x x. y ) e. { -u 1 , 1 } )
35		oveq2 6289	. . . . . . 7 |- ( x = -u 1 -> ( 1 / x ) = ( 1 / -u 1 ) )
36		ax-1ne0 9564	. . . . . . . . . 10 |- 1 =/= 0
37		divneg2 10274	. . . . . . . . . 10 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
38	5, 5, 36, 37	mp3an 1325	. . . . . . . . 9 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
39		1div1e1 10243	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	39	negeqi 9818	. . . . . . . . 9 |- -u ( 1 / 1 ) = -u 1
41	38, 40	eqtr3i 2474	. . . . . . . 8 |- ( 1 / -u 1 ) = -u 1
42	41, 2	eqeltri 2527	. . . . . . 7 |- ( 1 / -u 1 ) e. { -u 1 , 1 }
43	35, 42	syl6eqel 2539	. . . . . 6 |- ( x = -u 1 -> ( 1 / x ) e. { -u 1 , 1 } )
44		oveq2 6289	. . . . . . 7 |- ( x = 1 -> ( 1 / x ) = ( 1 / 1 ) )
45	39, 15	eqeltri 2527	. . . . . . 7 |- ( 1 / 1 ) e. { -u 1 , 1 }
46	44, 45	syl6eqel 2539	. . . . . 6 |- ( x = 1 -> ( 1 / x ) e. { -u 1 , 1 } )
47	43, 46	jaoi 379	. . . . 5 |- ( ( x = -u 1 \/ x = 1 ) -> ( 1 / x ) e. { -u 1 , 1 } )
48	8, 47	syl 16	. . . 4 |- ( x e. { -u 1 , 1 } -> ( 1 / x ) e. { -u 1 , 1 } )
49	48	adantr 465	. . 3 |- ( ( x e. { -u 1 , 1 } /\ x =/= 0 ) -> ( 1 / x ) e. { -u 1 , 1 } )
50	7, 34, 15, 49	expcl2lem 12157	. 2 |- ( ( -u 1 e. { -u 1 , 1 } /\ -u 1 =/= 0 /\ N e. ZZ ) -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
51	2, 3, 50	mp3an12 1315	1 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )

Syntax hints:   -> wi 4   \/ wo 368   = wceq 1383   e. wcel 1804   =/= wne 2638   C_ wss 3461   {cpr 4016  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496   x. cmul 9500   -ucneg 9811   / cdiv 10212   ZZcz 10870   ^cexp 12145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-n0 10802  df-z 10871  df-uz 11091  df-seq 12087  df-exp 12146

This theorem is referenced by:  m1expcl  12168  m1expaddsub  16397  psgnran  16414  psgnghm  18489  lgseisenlem2  23497  m1expevenALT  28536