Theorem m1expcl2 12007

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 9722	. . 3 |- -u 1 e. _V
2	1	prid1 4094	. 2 |- -u 1 e. { -u 1 , 1 }
3		neg1ne0 10541	. 2 |- -u 1 =/= 0
4		neg1cn 10539	. . . 4 |- -u 1 e. CC
5		ax-1cn 9454	. . . 4 |- 1 e. CC
6		prssi 4140	. . . 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 4008	. . . . 5 |- ( x e. { -u 1 , 1 } -> ( x = -u 1 \/ x = 1 ) )
9	7	sseli 3463	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> y e. CC )
10	9	mulm1d 9910	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) = -u y )
11		elpri 4008	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> ( y = -u 1 \/ y = 1 ) )
12		negeq 9716	. . . . . . . . . . 11 |- ( y = -u 1 -> -u y = -u -u 1 )
13		negneg1e1 10543	. . . . . . . . . . . 12 |- -u -u 1 = 1
14		1ex 9495	. . . . . . . . . . . . 13 |- 1 e. _V
15	14	prid2 4095	. . . . . . . . . . . 12 |- 1 e. { -u 1 , 1 }
16	13, 15	eqeltri 2538	. . . . . . . . . . 11 |- -u -u 1 e. { -u 1 , 1 }
17	12, 16	syl6eqel 2550	. . . . . . . . . 10 |- ( y = -u 1 -> -u y e. { -u 1 , 1 } )
18		negeq 9716	. . . . . . . . . . 11 |- ( y = 1 -> -u y = -u 1 )
19	18, 2	syl6eqel 2550	. . . . . . . . . 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 2542	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) e. { -u 1 , 1 } )
23		oveq1 6210	. . . . . . . 8 |- ( x = -u 1 -> ( x x. y ) = ( -u 1 x. y ) )
24	23	eleq1d 2523	. . . . . . 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 9518	. . . . . . . 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 2542	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( 1 x. y ) e. { -u 1 , 1 } )
29		oveq1 6210	. . . . . . . 8 |- ( x = 1 -> ( x x. y ) = ( 1 x. y ) )
30	29	eleq1d 2523	. . . . . . 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 6211	. . . . . . 7 |- ( x = -u 1 -> ( 1 / x ) = ( 1 / -u 1 ) )
36		ax-1ne0 9465	. . . . . . . . . 10 |- 1 =/= 0
37		divneg2 10169	. . . . . . . . . 10 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
38	5, 5, 36, 37	mp3an 1315	. . . . . . . . 9 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
39		1div1e1 10138	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	39	negeqi 9717	. . . . . . . . 9 |- -u ( 1 / 1 ) = -u 1
41	38, 40	eqtr3i 2485	. . . . . . . 8 |- ( 1 / -u 1 ) = -u 1
42	41, 2	eqeltri 2538	. . . . . . 7 |- ( 1 / -u 1 ) e. { -u 1 , 1 }
43	35, 42	syl6eqel 2550	. . . . . 6 |- ( x = -u 1 -> ( 1 / x ) e. { -u 1 , 1 } )
44		oveq2 6211	. . . . . . 7 |- ( x = 1 -> ( 1 / x ) = ( 1 / 1 ) )
45	39, 15	eqeltri 2538	. . . . . . 7 |- ( 1 / 1 ) e. { -u 1 , 1 }
46	44, 45	syl6eqel 2550	. . . . . 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 11997	. 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 1305	1 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )

Syntax hints:   -> wi 4   \/ wo 368   = wceq 1370   e. wcel 1758   =/= wne 2648   C_ wss 3439   {cpr 3990  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397   x. cmul 9401   -ucneg 9710   / cdiv 10107   ZZcz 10760   ^cexp 11985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-seq 11927  df-exp 11986

This theorem is referenced by:  m1expcl  12008  m1expaddsub  16126  psgnran  16143  psgnghm  18138  lgseisenlem2  22825  m1expevenALT  27271