Theorem dya2iocival 30335

Description: The function I returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 23368. (Contributed by Thierry Arnoux, 24-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 ) ) ) )

Assertion

Ref	Expression
dya2iocival	|- ( ( N e. ZZ /\ X e. ZZ ) -> ( X I N ) = ( ( X / ( 2 ^ N ) ) [,) ( ( X + 1 ) / ( 2 ^ N ) ) ) )

Distinct variable group:   x, n

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

Proof of Theorem dya2iocival

Dummy variables m u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( u = X -> ( u / ( 2 ^ m ) ) = ( X / ( 2 ^ m ) ) )
2		oveq1 6657	. . . . 5 |- ( u = X -> ( u + 1 ) = ( X + 1 ) )
3	2	oveq1d 6665	. . . 4 |- ( u = X -> ( ( u + 1 ) / ( 2 ^ m ) ) = ( ( X + 1 ) / ( 2 ^ m ) ) )
4	1, 3	oveq12d 6668	. . 3 |- ( u = X -> ( ( u / ( 2 ^ m ) ) [,) ( ( u + 1 ) / ( 2 ^ m ) ) ) = ( ( X / ( 2 ^ m ) ) [,) ( ( X + 1 ) / ( 2 ^ m ) ) ) )
5		oveq2 6658	. . . . 5 |- ( m = N -> ( 2 ^ m ) = ( 2 ^ N ) )
6	5	oveq2d 6666	. . . 4 |- ( m = N -> ( X / ( 2 ^ m ) ) = ( X / ( 2 ^ N ) ) )
7	5	oveq2d 6666	. . . 4 |- ( m = N -> ( ( X + 1 ) / ( 2 ^ m ) ) = ( ( X + 1 ) / ( 2 ^ N ) ) )
8	6, 7	oveq12d 6668	. . 3 |- ( m = N -> ( ( X / ( 2 ^ m ) ) [,) ( ( X + 1 ) / ( 2 ^ m ) ) ) = ( ( X / ( 2 ^ N ) ) [,) ( ( X + 1 ) / ( 2 ^ N ) ) ) )
9		dya2ioc.1	. . . 4 |- I = ( x e. ZZ , n e. ZZ |-> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
10		oveq1 6657	. . . . . 6 |- ( u = x -> ( u / ( 2 ^ m ) ) = ( x / ( 2 ^ m ) ) )
11		oveq1 6657	. . . . . . 7 |- ( u = x -> ( u + 1 ) = ( x + 1 ) )
12	11	oveq1d 6665	. . . . . 6 |- ( u = x -> ( ( u + 1 ) / ( 2 ^ m ) ) = ( ( x + 1 ) / ( 2 ^ m ) ) )
13	10, 12	oveq12d 6668	. . . . 5 |- ( u = x -> ( ( u / ( 2 ^ m ) ) [,) ( ( u + 1 ) / ( 2 ^ m ) ) ) = ( ( x / ( 2 ^ m ) ) [,) ( ( x + 1 ) / ( 2 ^ m ) ) ) )
14		oveq2 6658	. . . . . . 7 |- ( m = n -> ( 2 ^ m ) = ( 2 ^ n ) )
15	14	oveq2d 6666	. . . . . 6 |- ( m = n -> ( x / ( 2 ^ m ) ) = ( x / ( 2 ^ n ) ) )
16	14	oveq2d 6666	. . . . . 6 |- ( m = n -> ( ( x + 1 ) / ( 2 ^ m ) ) = ( ( x + 1 ) / ( 2 ^ n ) ) )
17	15, 16	oveq12d 6668	. . . . 5 |- ( m = n -> ( ( x / ( 2 ^ m ) ) [,) ( ( x + 1 ) / ( 2 ^ m ) ) ) = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
18	13, 17	cbvmpt2v 6735	. . . 4 |- ( u e. ZZ , m e. ZZ |-> ( ( u / ( 2 ^ m ) ) [,) ( ( u + 1 ) / ( 2 ^ m ) ) ) ) = ( x e. ZZ , n e. ZZ |-> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
19	9, 18	eqtr4i 2647	. . 3 |- I = ( u e. ZZ , m e. ZZ |-> ( ( u / ( 2 ^ m ) ) [,) ( ( u + 1 ) / ( 2 ^ m ) ) ) )
20		ovex 6678	. . 3 |- ( ( X / ( 2 ^ N ) ) [,) ( ( X + 1 ) / ( 2 ^ N ) ) ) e. _V
21	4, 8, 19, 20	ovmpt2 6796	. 2 |- ( ( X e. ZZ /\ N e. ZZ ) -> ( X I N ) = ( ( X / ( 2 ^ N ) ) [,) ( ( X + 1 ) / ( 2 ^ N ) ) ) )
22	21	ancoms 469	1 |- ( ( N e. ZZ /\ X e. ZZ ) -> ( X I N ) = ( ( X / ( 2 ^ N ) ) [,) ( ( X + 1 ) / ( 2 ^ N ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1c1 9937   + caddc 9939   / cdiv 10684   2c2 11070   ZZcz 11377   (,)cioo 12175   [,)cico 12177   ^cexp 12860   topGenctg 16098

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  dya2iocress  30336  dya2iocbrsiga  30337  dya2icoseg  30339