Theorem m1expcl2 10743

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 9176	. . 3 |- -u 1 e. CC
2		prid1g 3747	. . 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 9180	. 2 |- -u 1 # 0
5		ax-1cn 8053	. . . 4 |- 1 e. CC
6		prssi 3802	. . . 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 3666	. . . . 5 |- ( x e. { -u 1 , 1 } -> ( x = -u 1 \/ x = 1 ) )
9	7	sseli 3197	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> y e. CC )
10	9	mulm1d 8517	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) = -u y )
11		elpri 3666	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> ( y = -u 1 \/ y = 1 ) )
12		negeq 8300	. . . . . . . . . . 11 |- ( y = -u 1 -> -u y = -u -u 1 )
13		negneg1e1 9181	. . . . . . . . . . . 12 |- -u -u 1 = 1
14		1ex 8102	. . . . . . . . . . . . 13 |- 1 e. _V
15	14	prid2 3750	. . . . . . . . . . . 12 |- 1 e. { -u 1 , 1 }
16	13, 15	eqeltri 2280	. . . . . . . . . . 11 |- -u -u 1 e. { -u 1 , 1 }
17	12, 16	eqeltrdi 2298	. . . . . . . . . 10 |- ( y = -u 1 -> -u y e. { -u 1 , 1 } )
18		negeq 8300	. . . . . . . . . . 11 |- ( y = 1 -> -u y = -u 1 )
19	18, 3	eqeltrdi 2298	. . . . . . . . . 10 |- ( y = 1 -> -u y e. { -u 1 , 1 } )
20	17, 19	jaoi 718	. . . . . . . . 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 2284	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) e. { -u 1 , 1 } )
23		oveq1 5974	. . . . . . . 8 |- ( x = -u 1 -> ( x x. y ) = ( -u 1 x. y ) )
24	23	eleq1d 2276	. . . . . . 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 8126	. . . . . . . 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 2284	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( 1 x. y ) e. { -u 1 , 1 } )
29		oveq1 5974	. . . . . . . 8 |- ( x = 1 -> ( x x. y ) = ( 1 x. y ) )
30	29	eleq1d 2276	. . . . . . 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 718	. . . . 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 5975	. . . . . . 7 |- ( x = -u 1 -> ( 1 / x ) = ( 1 / -u 1 ) )
36		1ap0 8698	. . . . . . . . . 10 |- 1 # 0
37		divneg2ap 8844	. . . . . . . . . 10 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 # 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
38	5, 5, 36, 37	mp3an 1350	. . . . . . . . 9 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
39		1div1e1 8812	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	39	negeqi 8301	. . . . . . . . 9 |- -u ( 1 / 1 ) = -u 1
41	38, 40	eqtr3i 2230	. . . . . . . 8 |- ( 1 / -u 1 ) = -u 1
42	41, 3	eqeltri 2280	. . . . . . 7 |- ( 1 / -u 1 ) e. { -u 1 , 1 }
43	35, 42	eqeltrdi 2298	. . . . . 6 |- ( x = -u 1 -> ( 1 / x ) e. { -u 1 , 1 } )
44		oveq2 5975	. . . . . . 7 |- ( x = 1 -> ( 1 / x ) = ( 1 / 1 ) )
45	39, 15	eqeltri 2280	. . . . . . 7 |- ( 1 / 1 ) e. { -u 1 , 1 }
46	44, 45	eqeltrdi 2298	. . . . . 6 |- ( x = 1 -> ( 1 / x ) e. { -u 1 , 1 } )
47	43, 46	jaoi 718	. . . . 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 10733	. 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 1340	1 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2178   C_ wss 3174   {cpr 3644   class class class wbr 4059  (class class class)co 5967   CCcc 7958   0cc0 7960   1c1 7961   x. cmul 7965   -ucneg 8279   # cap 8689   / cdiv 8780   ZZcz 9407   ^cexp 10720

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  m1expcl  10744  m1expeven  10768  gausslemma2dlem0i  15649  lgseisenlem2  15663