Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iccordt | Structured version Visualization version GIF version |
Description: A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
iccordt | ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7234 | . 2 ⊢ (𝐴[,]𝐵) = ([,]‘〈𝐴, 𝐵〉) | |
2 | letsr 18123 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
3 | ledm 18120 | . . . . . . 7 ⊢ ℝ* = dom ≤ | |
4 | 3 | ordtcld3 22120 | . . . . . 6 ⊢ (( ≤ ∈ TosetRel ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
5 | 2, 4 | mp3an1 1450 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ ))) |
6 | 5 | rgen2 3125 | . . . 4 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) |
7 | df-icc 12966 | . . . . 5 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
8 | 7 | fmpo 7856 | . . . 4 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ∈ (Clsd‘(ordTop‘ ≤ )) ↔ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ ))) |
9 | 6, 8 | mpbi 233 | . . 3 ⊢ [,]:(ℝ* × ℝ*)⟶(Clsd‘(ordTop‘ ≤ )) |
10 | letop 22127 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top | |
11 | 0cld 21959 | . . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (Clsd‘(ordTop‘ ≤ ))) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ ∅ ∈ (Clsd‘(ordTop‘ ≤ )) |
13 | 9, 12 | f0cli 6935 | . 2 ⊢ ([,]‘〈𝐴, 𝐵〉) ∈ (Clsd‘(ordTop‘ ≤ )) |
14 | 1, 13 | eqeltri 2835 | 1 ⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2111 ∀wral 3062 {crab 3066 ∅c0 4251 〈cop 4561 class class class wbr 5067 × cxp 5563 ⟶wf 6393 ‘cfv 6397 (class class class)co 7231 ℝ*cxr 10890 ≤ cle 10892 [,]cicc 12962 ordTopcordt 17028 TosetRel ctsr 18095 Topctop 21814 Clsdccld 21937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-pre-lttri 10827 ax-pre-lttrn 10828 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-1o 8222 df-er 8411 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fi 9051 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-icc 12966 df-topgen 16972 df-ordt 17030 df-ps 18096 df-tsr 18097 df-top 21815 df-topon 21832 df-bases 21867 df-cld 21940 |
This theorem is referenced by: lecldbas 22140 icccldii 45913 |
Copyright terms: Public domain | W3C validator |