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Mirrors > Home > MPE Home > Th. List > Mathboxes > dya2iocbrsiga | Structured version Visualization version GIF version |
Description: Dyadic intervals are Borel sets of ℝ. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
Ref | Expression |
---|---|
sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
Ref | Expression |
---|---|
dya2iocbrsiga | ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ∈ 𝔅ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sxbrsiga.0 | . . 3 ⊢ 𝐽 = (topGen‘ran (,)) | |
2 | dya2ioc.1 | . . 3 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) | |
3 | 1, 2 | dya2iocival 32481 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
4 | mnfxr 11125 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → -∞ ∈ ℝ*) |
6 | simpr 485 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℤ) | |
7 | 6 | zred 12519 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℝ) |
8 | 2rp 12828 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 2 ∈ ℝ+) |
10 | simpl 483 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 𝑁 ∈ ℤ) | |
11 | 9, 10 | rpexpcld 14055 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (2↑𝑁) ∈ ℝ+) |
12 | 7, 11 | rerpdivcld 12896 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋 / (2↑𝑁)) ∈ ℝ) |
13 | 12 | rexrd 11118 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋 / (2↑𝑁)) ∈ ℝ*) |
14 | 1red 11069 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 1 ∈ ℝ) | |
15 | 7, 14 | readdcld 11097 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋 + 1) ∈ ℝ) |
16 | 15, 11 | rerpdivcld 12896 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) / (2↑𝑁)) ∈ ℝ) |
17 | 16 | rexrd 11118 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) / (2↑𝑁)) ∈ ℝ*) |
18 | mnflt 12952 | . . . . 5 ⊢ ((𝑋 / (2↑𝑁)) ∈ ℝ → -∞ < (𝑋 / (2↑𝑁))) | |
19 | 12, 18 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → -∞ < (𝑋 / (2↑𝑁))) |
20 | difioo 31331 | . . . 4 ⊢ (((-∞ ∈ ℝ* ∧ (𝑋 / (2↑𝑁)) ∈ ℝ* ∧ ((𝑋 + 1) / (2↑𝑁)) ∈ ℝ*) ∧ -∞ < (𝑋 / (2↑𝑁))) → ((-∞(,)((𝑋 + 1) / (2↑𝑁))) ∖ (-∞(,)(𝑋 / (2↑𝑁)))) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) | |
21 | 5, 13, 17, 19, 20 | syl31anc 1372 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((-∞(,)((𝑋 + 1) / (2↑𝑁))) ∖ (-∞(,)(𝑋 / (2↑𝑁)))) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
22 | brsigarn 32391 | . . . . 5 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
23 | elrnsiga 32333 | . . . . 5 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
24 | 22, 23 | ax-mp 5 | . . . 4 ⊢ 𝔅ℝ ∈ ∪ ran sigAlgebra |
25 | retop 24023 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
26 | iooretop 24027 | . . . . . 6 ⊢ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (topGen‘ran (,)) | |
27 | elsigagen 32354 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (topGen‘ran (,))) → (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,)))) | |
28 | 25, 26, 27 | mp2an 689 | . . . . 5 ⊢ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,))) |
29 | df-brsiga 32389 | . . . . 5 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
30 | 28, 29 | eleqtrri 2836 | . . . 4 ⊢ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ 𝔅ℝ |
31 | iooretop 24027 | . . . . . 6 ⊢ (-∞(,)(𝑋 / (2↑𝑁))) ∈ (topGen‘ran (,)) | |
32 | elsigagen 32354 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)(𝑋 / (2↑𝑁))) ∈ (topGen‘ran (,))) → (-∞(,)(𝑋 / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,)))) | |
33 | 25, 31, 32 | mp2an 689 | . . . . 5 ⊢ (-∞(,)(𝑋 / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,))) |
34 | 33, 29 | eleqtrri 2836 | . . . 4 ⊢ (-∞(,)(𝑋 / (2↑𝑁))) ∈ 𝔅ℝ |
35 | difelsiga 32340 | . . . 4 ⊢ ((𝔅ℝ ∈ ∪ ran sigAlgebra ∧ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ 𝔅ℝ ∧ (-∞(,)(𝑋 / (2↑𝑁))) ∈ 𝔅ℝ) → ((-∞(,)((𝑋 + 1) / (2↑𝑁))) ∖ (-∞(,)(𝑋 / (2↑𝑁)))) ∈ 𝔅ℝ) | |
36 | 24, 30, 34, 35 | mp3an 1460 | . . 3 ⊢ ((-∞(,)((𝑋 + 1) / (2↑𝑁))) ∖ (-∞(,)(𝑋 / (2↑𝑁)))) ∈ 𝔅ℝ |
37 | 21, 36 | eqeltrrdi 2846 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ 𝔅ℝ) |
38 | 3, 37 | eqeltrd 2837 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ∈ 𝔅ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∖ cdif 3894 ∪ cuni 4851 class class class wbr 5089 ran crn 5615 ‘cfv 6473 (class class class)co 7329 ∈ cmpo 7331 ℝcr 10963 1c1 10965 + caddc 10967 -∞cmnf 11100 ℝ*cxr 11101 < clt 11102 / cdiv 11725 2c2 12121 ℤcz 12412 ℝ+crp 12823 (,)cioo 13172 [,)cico 13174 ↑cexp 13875 topGenctg 17237 Topctop 22140 sigAlgebracsiga 32315 sigaGencsigagen 32345 𝔅ℝcbrsiga 32388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-ac2 10312 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-2o 8360 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-sup 9291 df-inf 9292 df-oi 9359 df-dju 9750 df-card 9788 df-acn 9791 df-ac 9965 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-n0 12327 df-z 12413 df-uz 12676 df-q 12782 df-rp 12824 df-ioo 13176 df-ico 13178 df-seq 13815 df-exp 13876 df-topgen 17243 df-top 22141 df-bases 22194 df-siga 32316 df-sigagen 32346 df-brsiga 32389 |
This theorem is referenced by: (None) |
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