Theorem m1expcl2 10346

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 8849	. . 3 |- -u 1 e. CC
2		prid1g 3635	. . 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 8853	. 2 |- -u 1 # 0
5		ax-1cn 7737	. . . 4 |- 1 e. CC
6		prssi 3686	. . . 4 |- ( ( -u 1 e. CC /\ 1 e. CC ) -> { -u 1 , 1 } C_ CC )
7	1, 5, 6	mp2an 423	. . 3 |- { -u 1 , 1 } C_ CC
8		elpri 3555	. . . . 5 |- ( x e. { -u 1 , 1 } -> ( x = -u 1 \/ x = 1 ) )
9	7	sseli 3098	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> y e. CC )
10	9	mulm1d 8196	. . . . . . . 8 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) = -u y )
11		elpri 3555	. . . . . . . . 9 |- ( y e. { -u 1 , 1 } -> ( y = -u 1 \/ y = 1 ) )
12		negeq 7979	. . . . . . . . . . 11 |- ( y = -u 1 -> -u y = -u -u 1 )
13		negneg1e1 8854	. . . . . . . . . . . 12 |- -u -u 1 = 1
14		1ex 7785	. . . . . . . . . . . . 13 |- 1 e. _V
15	14	prid2 3638	. . . . . . . . . . . 12 |- 1 e. { -u 1 , 1 }
16	13, 15	eqeltri 2213	. . . . . . . . . . 11 |- -u -u 1 e. { -u 1 , 1 }
17	12, 16	eqeltrdi 2231	. . . . . . . . . 10 |- ( y = -u 1 -> -u y e. { -u 1 , 1 } )
18		negeq 7979	. . . . . . . . . . 11 |- ( y = 1 -> -u y = -u 1 )
19	18, 3	eqeltrdi 2231	. . . . . . . . . 10 |- ( y = 1 -> -u y e. { -u 1 , 1 } )
20	17, 19	jaoi 706	. . . . . . . . 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 2217	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( -u 1 x. y ) e. { -u 1 , 1 } )
23		oveq1 5789	. . . . . . . 8 |- ( x = -u 1 -> ( x x. y ) = ( -u 1 x. y ) )
24	23	eleq1d 2209	. . . . . . 7 |- ( x = -u 1 -> ( ( x x. y ) e. { -u 1 , 1 } <-> ( -u 1 x. y ) e. { -u 1 , 1 } ) )
25	22, 24	syl5ibr 155	. . . . . 6 |- ( x = -u 1 -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
26	9	mulid2d 7808	. . . . . . . 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 2217	. . . . . . 7 |- ( y e. { -u 1 , 1 } -> ( 1 x. y ) e. { -u 1 , 1 } )
29		oveq1 5789	. . . . . . . 8 |- ( x = 1 -> ( x x. y ) = ( 1 x. y ) )
30	29	eleq1d 2209	. . . . . . 7 |- ( x = 1 -> ( ( x x. y ) e. { -u 1 , 1 } <-> ( 1 x. y ) e. { -u 1 , 1 } ) )
31	28, 30	syl5ibr 155	. . . . . 6 |- ( x = 1 -> ( y e. { -u 1 , 1 } -> ( x x. y ) e. { -u 1 , 1 } ) )
32	25, 31	jaoi 706	. . . . 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 123	. . 3 |- ( ( x e. { -u 1 , 1 } /\ y e. { -u 1 , 1 } ) -> ( x x. y ) e. { -u 1 , 1 } )
35		oveq2 5790	. . . . . . 7 |- ( x = -u 1 -> ( 1 / x ) = ( 1 / -u 1 ) )
36		1ap0 8376	. . . . . . . . . 10 |- 1 # 0
37		divneg2ap 8520	. . . . . . . . . 10 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 # 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
38	5, 5, 36, 37	mp3an 1316	. . . . . . . . 9 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
39		1div1e1 8488	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	39	negeqi 7980	. . . . . . . . 9 |- -u ( 1 / 1 ) = -u 1
41	38, 40	eqtr3i 2163	. . . . . . . 8 |- ( 1 / -u 1 ) = -u 1
42	41, 3	eqeltri 2213	. . . . . . 7 |- ( 1 / -u 1 ) e. { -u 1 , 1 }
43	35, 42	eqeltrdi 2231	. . . . . 6 |- ( x = -u 1 -> ( 1 / x ) e. { -u 1 , 1 } )
44		oveq2 5790	. . . . . . 7 |- ( x = 1 -> ( 1 / x ) = ( 1 / 1 ) )
45	39, 15	eqeltri 2213	. . . . . . 7 |- ( 1 / 1 ) e. { -u 1 , 1 }
46	44, 45	eqeltrdi 2231	. . . . . 6 |- ( x = 1 -> ( 1 / x ) e. { -u 1 , 1 } )
47	43, 46	jaoi 706	. . . . 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 274	. . 3 |- ( ( x e. { -u 1 , 1 } /\ x # 0 ) -> ( 1 / x ) e. { -u 1 , 1 } )
50	7, 34, 15, 49	expcl2lemap 10336	. 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 1306	1 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1332   e. wcel 1481   C_ wss 3076   {cpr 3533   class class class wbr 3937  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   x. cmul 7649   -ucneg 7958   # cap 8367   / cdiv 8456   ZZcz 9078   ^cexp 10323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250  df-exp 10324

This theorem is referenced by:  m1expcl  10347  m1expeven  10371