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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 31531 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
4 | mnfxr 10698 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → -∞ ∈ ℝ*) |
6 | simpr 487 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℤ) | |
7 | 6 | zred 12088 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 𝑋 ∈ ℝ) |
8 | 2rp 12395 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 2 ∈ ℝ+) |
10 | simpl 485 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 𝑁 ∈ ℤ) | |
11 | 9, 10 | rpexpcld 13609 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (2↑𝑁) ∈ ℝ+) |
12 | 7, 11 | rerpdivcld 12463 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋 / (2↑𝑁)) ∈ ℝ) |
13 | 12 | rexrd 10691 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋 / (2↑𝑁)) ∈ ℝ*) |
14 | 1red 10642 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → 1 ∈ ℝ) | |
15 | 7, 14 | readdcld 10670 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋 + 1) ∈ ℝ) |
16 | 15, 11 | rerpdivcld 12463 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) / (2↑𝑁)) ∈ ℝ) |
17 | 16 | rexrd 10691 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((𝑋 + 1) / (2↑𝑁)) ∈ ℝ*) |
18 | mnflt 12519 | . . . . 5 ⊢ ((𝑋 / (2↑𝑁)) ∈ ℝ → -∞ < (𝑋 / (2↑𝑁))) | |
19 | 12, 18 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → -∞ < (𝑋 / (2↑𝑁))) |
20 | difioo 30505 | . . . 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 1369 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((-∞(,)((𝑋 + 1) / (2↑𝑁))) ∖ (-∞(,)(𝑋 / (2↑𝑁)))) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
22 | brsigarn 31443 | . . . . 5 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
23 | elrnsiga 31385 | . . . . 5 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
24 | 22, 23 | ax-mp 5 | . . . 4 ⊢ 𝔅ℝ ∈ ∪ ran sigAlgebra |
25 | retop 23370 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
26 | iooretop 23374 | . . . . . 6 ⊢ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (topGen‘ran (,)) | |
27 | elsigagen 31406 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (topGen‘ran (,))) → (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,)))) | |
28 | 25, 26, 27 | mp2an 690 | . . . . 5 ⊢ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,))) |
29 | df-brsiga 31441 | . . . . 5 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
30 | 28, 29 | eleqtrri 2912 | . . . 4 ⊢ (-∞(,)((𝑋 + 1) / (2↑𝑁))) ∈ 𝔅ℝ |
31 | iooretop 23374 | . . . . . 6 ⊢ (-∞(,)(𝑋 / (2↑𝑁))) ∈ (topGen‘ran (,)) | |
32 | elsigagen 31406 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)(𝑋 / (2↑𝑁))) ∈ (topGen‘ran (,))) → (-∞(,)(𝑋 / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,)))) | |
33 | 25, 31, 32 | mp2an 690 | . . . . 5 ⊢ (-∞(,)(𝑋 / (2↑𝑁))) ∈ (sigaGen‘(topGen‘ran (,))) |
34 | 33, 29 | eleqtrri 2912 | . . . 4 ⊢ (-∞(,)(𝑋 / (2↑𝑁))) ∈ 𝔅ℝ |
35 | difelsiga 31392 | . . . 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 1457 | . . 3 ⊢ ((-∞(,)((𝑋 + 1) / (2↑𝑁))) ∖ (-∞(,)(𝑋 / (2↑𝑁)))) ∈ 𝔅ℝ |
37 | 21, 36 | eqeltrrdi 2922 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ 𝔅ℝ) |
38 | 3, 37 | eqeltrd 2913 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ∈ 𝔅ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ∪ cuni 4838 class class class wbr 5066 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ℝcr 10536 1c1 10538 + caddc 10540 -∞cmnf 10673 ℝ*cxr 10674 < clt 10675 / cdiv 11297 2c2 11693 ℤcz 11982 ℝ+crp 12390 (,)cioo 12739 [,)cico 12741 ↑cexp 13430 topGenctg 16711 Topctop 21501 sigAlgebracsiga 31367 sigaGencsigagen 31397 𝔅ℝcbrsiga 31440 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-ioo 12743 df-ico 12745 df-seq 13371 df-exp 13431 df-topgen 16717 df-top 21502 df-bases 21554 df-siga 31368 df-sigagen 31398 df-brsiga 31441 |
This theorem is referenced by: (None) |
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