Theorem m1expcl2 10635

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		neg1cn 9089	. . 3 |- -u 1 e. CC
2		prid1g 3723	. . 3 |- ( -u 1 e. CC -> -u 1 e. { -u 1 , 1 } )
3	1, 2	ax-mp 5	. 2 |- -u 1 e. { -u 1 , 1 }
4		neg1ap0 9093	. 2 |- -u 1 # 0
5		ax-1cn 7967	. . . 4 |- 1 e. CC
6		prssi 3777	. . . 4 |- ( ( -u 1 e. CC /\ 1 e. CC ) -> { -u 1 , 1 } C_ CC )
7	1, 5, 6	mp2an 426	. . 3 |- { -u 1 , 1 } C_ CC
8		elpri 3642	. . . . 5 |- ( x e. { -u 1 , 1 } -> ( x = -u 1 \/ x = 1 ) )
9	7	sseli 3176	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> y e. CC )
10	9	mulm1d 8431	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) = -u y )
11		elpri 3642	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> ( y = -u 1 \/ y = 1 ) )
12		negeq 8214	. . . . . . . . . . 11 |- ( y = -u 1 -> -u y = -u -u 1 )
13		negneg1e1 9094	. . . . . . . . . . . 12 |- -u -u 1 = 1
14		1ex 8016	. . . . . . . . . . . . 13 |- 1 e. _V
15	14	prid2 3726	. . . . . . . . . . . 12 |- 1 e. { -u 1 , 1 }
16	13, 15	eqeltri 2266	. . . . . . . . . . 11 |- -u -u 1 e. { -u 1 , 1 }
17	12, 16	eqeltrdi 2284	. . . . . . . . . 10 |- ( y = -u 1 -> -u y e. { -u 1 , 1 } )
18		negeq 8214	. . . . . . . . . . 11 |- ( y = 1 -> -u y = -u 1 )
19	18, 3	eqeltrdi 2284	. . . . . . . . . 10 |- ( y = 1 -> -u y e. { -u 1 , 1 } )
20	17, 19	jaoi 717	. . . . . . . . 9 |- ( ( y = -u 1 \/ y = 1 ) -> -u y e. { -u 1 , 1 } )
21	11, 20	syl 14	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> -u y e. { -u 1 , 1 } )
22	10, 21	eqeltrd 2270	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) e. { -u 1 , 1 } )
23		oveq1 5926	. . . . . . . 8 |- ( x = -u 1 -> ( x x. y ) = ( -u 1 x. y ) )
24	23	eleq1d 2262	. . . . . . 7 |- ( x = -u 1 -> ( ( x x. y ) e. { -u 1 , 1 } <-> ( -u 1 x. y ) e. { -u 1 , 1 } ) )
25	22, 24	imbitrrid 156	. . . . . 6 |- ( x = -u 1 -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
26	9	mulid2d 8040	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> ( 1 x. y ) = y )
27		id 19	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> y e. { -u 1 , 1 } )
28	26, 27	eqeltrd 2270	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( 1 x. y ) e. { -u 1 , 1 } )
29		oveq1 5926	. . . . . . . 8 |- ( x = 1 -> ( x x. y ) = ( 1 x. y ) )
30	29	eleq1d 2262	. . . . . . 7 |- ( x = 1 -> ( ( x x. y ) e. { -u 1 , 1 } <-> ( 1 x. y ) e. { -u 1 , 1 } ) )
31	28, 30	imbitrrid 156	. . . . . 6 |- ( x = 1 -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
32	25, 31	jaoi 717	. . . . 5 |- ( ( x = -u 1 \/ x = 1 ) -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
33	8, 32	syl 14	. . . 4 |- ( x e. { -u 1 , 1 } -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
34	33	imp 124	. . 3 |- ( ( x e. { -u 1 , 1 } /\ y e. { -u 1 , 1 } ) -> ( x x. y ) e. { -u 1 , 1 } )
35		oveq2 5927	. . . . . . 7 |- ( x = -u 1 -> ( 1 / x ) = ( 1 / -u 1 ) )
36		1ap0 8611	. . . . . . . . . 10 |- 1 # 0
37		divneg2ap 8757	. . . . . . . . . 10 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 # 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
38	5, 5, 36, 37	mp3an 1348	. . . . . . . . 9 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
39		1div1e1 8725	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	39	negeqi 8215	. . . . . . . . 9 |- -u ( 1 / 1 ) = -u 1
41	38, 40	eqtr3i 2216	. . . . . . . 8 |- ( 1 / -u 1 ) = -u 1
42	41, 3	eqeltri 2266	. . . . . . 7 |- ( 1 / -u 1 ) e. { -u 1 , 1 }
43	35, 42	eqeltrdi 2284	. . . . . 6 |- ( x = -u 1 -> ( 1 / x ) e. { -u 1 , 1 } )
44		oveq2 5927	. . . . . . 7 |- ( x = 1 -> ( 1 / x ) = ( 1 / 1 ) )
45	39, 15	eqeltri 2266	. . . . . . 7 |- ( 1 / 1 ) e. { -u 1 , 1 }
46	44, 45	eqeltrdi 2284	. . . . . 6 |- ( x = 1 -> ( 1 / x ) e. { -u 1 , 1 } )
47	43, 46	jaoi 717	. . . . 5 |- ( ( x = -u 1 \/ x = 1 ) -> ( 1 / x ) e. { -u 1 , 1 } )
48	8, 47	syl 14	. . . 4 |- ( x e. { -u 1 , 1 } -> ( 1 / x ) e. { -u 1 , 1 } )
49	48	adantr 276	. . 3 |- ( ( x e. { -u 1 , 1 } /\ x # 0 ) -> ( 1 / x ) e. { -u 1 , 1 } )
50	7, 34, 15, 49	expcl2lemap 10625	. 2 |- ( ( -u 1 e. { -u 1 , 1 } /\ -u 1 # 0 /\ N e. ZZ ) -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
51	3, 4, 50	mp3an12 1338	1 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2164   C_ wss 3154   {cpr 3620   class class class wbr 4030  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   x. cmul 7879   -ucneg 8193   # cap 8602   / cdiv 8693   ZZcz 9320   ^cexp 10612

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522  df-exp 10613

This theorem is referenced by:  m1expcl  10636  m1expeven  10660  gausslemma2dlem0i  15214  lgseisenlem2  15228