Theorem dya2icoseg 30339

Description: For any point and any closed-below, open-above interval of RR centered on that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 19-Sep-2017.)

Hypotheses

Ref	Expression
sxbrsiga.0	|- J = ( topGen ` ran (,) )
dya2ioc.1	|- I = ( x e. ZZ , n e. ZZ |-> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
dya2icoseg.1	|- N = ( |_ ` ( 1 - ( 2 logb D ) ) )

Assertion

Ref	Expression
dya2icoseg	|- ( ( X e. RR /\ D e. RR+ ) -> E. b e. ran I ( X e. b /\ b C_ ( ( X - D ) (,) ( X + D ) ) ) )

Distinct variable groups:   x, n   x, I   D, b   I, b, x   N, b, x   X, b, x

Allowed substitution hints:   D( x, n)   I( n)   J( x, n, b)   N( n)   X( n)

Proof of Theorem dya2icoseg

Step	Hyp	Ref	Expression
1		dya2ioc.1	. . . . 5 |- I = ( x e. ZZ , n e. ZZ |-> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
2		ovex 6678	. . . . 5 |- ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. _V
3	1, 2	fnmpt2i 7239	. . . 4 |- I Fn ( ZZ X. ZZ )
4	3	a1i 11	. . 3 |- ( ( X e. RR /\ D e. RR+ ) -> I Fn ( ZZ X. ZZ ) )
5		simpl 473	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> X e. RR )
6		2rp 11837	. . . . . . 7 |- 2 e. RR+
7		dya2icoseg.1	. . . . . . . 8 |- N = ( |_ ` ( 1 - ( 2 logb D ) ) )
8		1red 10055	. . . . . . . . . 10 |- ( D e. RR+ -> 1 e. RR )
9		2z 11409	. . . . . . . . . . . 12 |- 2 e. ZZ
10		uzid 11702	. . . . . . . . . . . 12 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
11	9, 10	ax-mp 5	. . . . . . . . . . 11 |- 2 e. ( ZZ>= ` 2 )
12		relogbzcl 24512	. . . . . . . . . . 11 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ D e. RR+ ) -> ( 2 logb D ) e. RR )
13	11, 12	mpan 706	. . . . . . . . . 10 |- ( D e. RR+ -> ( 2 logb D ) e. RR )
14	8, 13	resubcld 10458	. . . . . . . . 9 |- ( D e. RR+ -> ( 1 - ( 2 logb D ) ) e. RR )
15	14	flcld 12599	. . . . . . . 8 |- ( D e. RR+ -> ( |_ ` ( 1 - ( 2 logb D ) ) ) e. ZZ )
16	7, 15	syl5eqel 2705	. . . . . . 7 |- ( D e. RR+ -> N e. ZZ )
17		rpexpcl 12879	. . . . . . . 8 |- ( ( 2 e. RR+ /\ N e. ZZ ) -> ( 2 ^ N ) e. RR+ )
18	17	rpred 11872	. . . . . . 7 |- ( ( 2 e. RR+ /\ N e. ZZ ) -> ( 2 ^ N ) e. RR )
19	6, 16, 18	sylancr 695	. . . . . 6 |- ( D e. RR+ -> ( 2 ^ N ) e. RR )
20	19	adantl 482	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( 2 ^ N ) e. RR )
21	5, 20	remulcld 10070	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> ( X x. ( 2 ^ N ) ) e. RR )
22	21	flcld 12599	. . 3 |- ( ( X e. RR /\ D e. RR+ ) -> ( |_ ` ( X x. ( 2 ^ N ) ) ) e. ZZ )
23	16	adantl 482	. . 3 |- ( ( X e. RR /\ D e. RR+ ) -> N e. ZZ )
24		fnovrn 6809	. . 3 |- ( ( I Fn ( ZZ X. ZZ ) /\ ( |_ ` ( X x. ( 2 ^ N ) ) ) e. ZZ /\ N e. ZZ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) e. ran I )
25	4, 22, 23, 24	syl3anc 1326	. 2 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) e. ran I )
26	22	zred 11482	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( |_ ` ( X x. ( 2 ^ N ) ) ) e. RR )
27	6, 23, 17	sylancr 695	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( 2 ^ N ) e. RR+ )
28		fllelt 12598	. . . . . . . 8 |- ( ( X x. ( 2 ^ N ) ) e. RR -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) <_ ( X x. ( 2 ^ N ) ) /\ ( X x. ( 2 ^ N ) ) < ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) ) )
29	21, 28	syl 17	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) <_ ( X x. ( 2 ^ N ) ) /\ ( X x. ( 2 ^ N ) ) < ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) ) )
30	29	simpld 475	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( |_ ` ( X x. ( 2 ^ N ) ) ) <_ ( X x. ( 2 ^ N ) ) )
31	26, 21, 27, 30	lediv1dd 11930	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) <_ ( ( X x. ( 2 ^ N ) ) / ( 2 ^ N ) ) )
32	5	recnd 10068	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> X e. CC )
33	20	recnd 10068	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( 2 ^ N ) e. CC )
34		2cnd 11093	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> 2 e. CC )
35		2ne0 11113	. . . . . . . 8 |- 2 =/= 0
36	35	a1i 11	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> 2 =/= 0 )
37	34, 36, 23	expne0d 13014	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( 2 ^ N ) =/= 0 )
38	32, 33, 37	divcan4d 10807	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( X x. ( 2 ^ N ) ) / ( 2 ^ N ) ) = X )
39	31, 38	breqtrd 4679	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) <_ X )
40		1red 10055	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> 1 e. RR )
41	26, 40	readdcld 10069	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) e. RR )
42	29	simprd 479	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( X x. ( 2 ^ N ) ) < ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) )
43	21, 41, 27, 42	ltdiv1dd 11929	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( X x. ( 2 ^ N ) ) / ( 2 ^ N ) ) < ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) )
44	38, 43	eqbrtrrd 4677	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> X < ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) )
45	26, 20, 37	redivcld 10853	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) e. RR )
46	41, 20, 37	redivcld 10853	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) e. RR )
47	46	rexrd 10089	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) e. RR* )
48		elico2 12237	. . . . 5 |- ( ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) e. RR /\ ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) e. RR* ) -> ( X e. ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) [,) ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) <-> ( X e. RR /\ ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) <_ X /\ X < ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) ) )
49	45, 47, 48	syl2anc 693	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> ( X e. ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) [,) ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) <-> ( X e. RR /\ ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) <_ X /\ X < ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) ) )
50	5, 39, 44, 49	mpbir3and 1245	. . 3 |- ( ( X e. RR /\ D e. RR+ ) -> X e. ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) [,) ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) )
51		sxbrsiga.0	. . . . 5 |- J = ( topGen ` ran (,) )
52	51, 1	dya2iocival 30335	. . . 4 |- ( ( N e. ZZ /\ ( |_ ` ( X x. ( 2 ^ N ) ) ) e. ZZ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) = ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) [,) ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) )
53	23, 22, 52	syl2anc 693	. . 3 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) = ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) [,) ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) )
54	50, 53	eleqtrrd 2704	. 2 |- ( ( X e. RR /\ D e. RR+ ) -> X e. ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) )
55		simpr 477	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> D e. RR+ )
56	55	rpred 11872	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> D e. RR )
57	5, 56	resubcld 10458	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( X - D ) e. RR )
58	57	rexrd 10089	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> ( X - D ) e. RR* )
59	5, 56	readdcld 10069	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( X + D ) e. RR )
60	59	rexrd 10089	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> ( X + D ) e. RR* )
61	20, 37	rereccld 10852	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( 1 / ( 2 ^ N ) ) e. RR )
62	5, 61	resubcld 10458	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( X - ( 1 / ( 2 ^ N ) ) ) e. RR )
63	7	oveq2i 6661	. . . . . . . 8 |- ( 2 ^ N ) = ( 2 ^ ( |_ ` ( 1 - ( 2 logb D ) ) ) )
64	63	oveq2i 6661	. . . . . . 7 |- ( 1 / ( 2 ^ N ) ) = ( 1 / ( 2 ^ ( |_ ` ( 1 - ( 2 logb D ) ) ) ) )
65		dya2ub 30332	. . . . . . . 8 |- ( D e. RR+ -> ( 1 / ( 2 ^ ( |_ ` ( 1 - ( 2 logb D ) ) ) ) ) < D )
66	65	adantl 482	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( 1 / ( 2 ^ ( |_ ` ( 1 - ( 2 logb D ) ) ) ) ) < D )
67	64, 66	syl5eqbr 4688	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( 1 / ( 2 ^ N ) ) < D )
68	61, 56, 5, 67	ltsub2dd 10640	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( X - D ) < ( X - ( 1 / ( 2 ^ N ) ) ) )
69	32, 33	mulcld 10060	. . . . . . . 8 |- ( ( X e. RR /\ D e. RR+ ) -> ( X x. ( 2 ^ N ) ) e. CC )
70		1cnd 10056	. . . . . . . 8 |- ( ( X e. RR /\ D e. RR+ ) -> 1 e. CC )
71	69, 70, 33, 37	divsubdird 10840	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( X x. ( 2 ^ N ) ) - 1 ) / ( 2 ^ N ) ) = ( ( ( X x. ( 2 ^ N ) ) / ( 2 ^ N ) ) - ( 1 / ( 2 ^ N ) ) ) )
72	38	oveq1d 6665	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( X x. ( 2 ^ N ) ) / ( 2 ^ N ) ) - ( 1 / ( 2 ^ N ) ) ) = ( X - ( 1 / ( 2 ^ N ) ) ) )
73	71, 72	eqtrd 2656	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( X x. ( 2 ^ N ) ) - 1 ) / ( 2 ^ N ) ) = ( X - ( 1 / ( 2 ^ N ) ) ) )
74	21, 40	resubcld 10458	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( X x. ( 2 ^ N ) ) - 1 ) e. RR )
75	21, 41, 40, 42	ltsub1dd 10639	. . . . . . . . 9 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( X x. ( 2 ^ N ) ) - 1 ) < ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) - 1 ) )
76	26	recnd 10068	. . . . . . . . . 10 |- ( ( X e. RR /\ D e. RR+ ) -> ( |_ ` ( X x. ( 2 ^ N ) ) ) e. CC )
77	76, 70	pncand 10393	. . . . . . . . 9 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) - 1 ) = ( |_ ` ( X x. ( 2 ^ N ) ) ) )
78	75, 77	breqtrd 4679	. . . . . . . 8 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( X x. ( 2 ^ N ) ) - 1 ) < ( |_ ` ( X x. ( 2 ^ N ) ) ) )
79	74, 26, 78	ltled 10185	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( X x. ( 2 ^ N ) ) - 1 ) <_ ( |_ ` ( X x. ( 2 ^ N ) ) ) )
80	74, 26, 27, 79	lediv1dd 11930	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( X x. ( 2 ^ N ) ) - 1 ) / ( 2 ^ N ) ) <_ ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) )
81	73, 80	eqbrtrrd 4677	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( X - ( 1 / ( 2 ^ N ) ) ) <_ ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) )
82	57, 62, 45, 68, 81	ltletrd 10197	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> ( X - D ) < ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) )
83	5, 61	readdcld 10069	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( X + ( 1 / ( 2 ^ N ) ) ) e. RR )
84	21, 40	readdcld 10069	. . . . . . . 8 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( X x. ( 2 ^ N ) ) + 1 ) e. RR )
85	26, 21, 40, 30	leadd1dd 10641	. . . . . . . 8 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) <_ ( ( X x. ( 2 ^ N ) ) + 1 ) )
86	41, 84, 27, 85	lediv1dd 11930	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) <_ ( ( ( X x. ( 2 ^ N ) ) + 1 ) / ( 2 ^ N ) ) )
87	69, 70, 33, 37	divdird 10839	. . . . . . . 8 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( X x. ( 2 ^ N ) ) + 1 ) / ( 2 ^ N ) ) = ( ( ( X x. ( 2 ^ N ) ) / ( 2 ^ N ) ) + ( 1 / ( 2 ^ N ) ) ) )
88	38	oveq1d 6665	. . . . . . . 8 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( X x. ( 2 ^ N ) ) / ( 2 ^ N ) ) + ( 1 / ( 2 ^ N ) ) ) = ( X + ( 1 / ( 2 ^ N ) ) ) )
89	87, 88	eqtrd 2656	. . . . . . 7 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( X x. ( 2 ^ N ) ) + 1 ) / ( 2 ^ N ) ) = ( X + ( 1 / ( 2 ^ N ) ) ) )
90	86, 89	breqtrd 4679	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) <_ ( X + ( 1 / ( 2 ^ N ) ) ) )
91	61, 56, 5, 67	ltadd2dd 10196	. . . . . 6 |- ( ( X e. RR /\ D e. RR+ ) -> ( X + ( 1 / ( 2 ^ N ) ) ) < ( X + D ) )
92	46, 83, 59, 90, 91	lelttrd 10195	. . . . 5 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) < ( X + D ) )
93	46, 59, 92	ltled 10185	. . . 4 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) <_ ( X + D ) )
94		icossioo 12264	. . . 4 |- ( ( ( ( X - D ) e. RR* /\ ( X + D ) e. RR* ) /\ ( ( X - D ) < ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) /\ ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) <_ ( X + D ) ) ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) [,) ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) C_ ( ( X - D ) (,) ( X + D ) ) )
95	58, 60, 82, 93, 94	syl22anc 1327	. . 3 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) / ( 2 ^ N ) ) [,) ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) + 1 ) / ( 2 ^ N ) ) ) C_ ( ( X - D ) (,) ( X + D ) ) )
96	53, 95	eqsstrd 3639	. 2 |- ( ( X e. RR /\ D e. RR+ ) -> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) C_ ( ( X - D ) (,) ( X + D ) ) )
97		eleq2 2690	. . . 4 |- ( b = ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) -> ( X e. b <-> X e. ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) ) )
98		sseq1 3626	. . . 4 |- ( b = ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) -> ( b C_ ( ( X - D ) (,) ( X + D ) ) <-> ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) C_ ( ( X - D ) (,) ( X + D ) ) ) )
99	97, 98	anbi12d 747	. . 3 |- ( b = ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) -> ( ( X e. b /\ b C_ ( ( X - D ) (,) ( X + D ) ) ) <-> ( X e. ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) /\ ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) C_ ( ( X - D ) (,) ( X + D ) ) ) ) )
100	99	rspcev 3309	. 2 |- ( ( ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) e. ran I /\ ( X e. ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) /\ ( ( |_ ` ( X x. ( 2 ^ N ) ) ) I N ) C_ ( ( X - D ) (,) ( X + D ) ) ) ) -> E. b e. ran I ( X e. b /\ b C_ ( ( X - D ) (,) ( X + D ) ) ) )
101	25, 54, 96, 100	syl12anc 1324	1 |- ( ( X e. RR /\ D e. RR+ ) -> E. b e. ran I ( X e. b /\ b C_ ( ( X - D ) (,) ( X + D ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   class class class wbr 4653   X. cxp 5112   ran crn 5115   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   (,)cioo 12175   [,)cico 12177   |_cfl 12591   ^cexp 12860   topGenctg 16098   log_b clogb 24502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-logb 24503

This theorem is referenced by:  dya2icoseg2  30340