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Mirrors > Home > MPE Home > Th. List > icopnfcld | Structured version Visualization version GIF version |
Description: Right-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
icopnfcld | ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 10096 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ*) |
3 | rexr 10085 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 10092 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
6 | mnflt 11957 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
7 | ltpnf 11954 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
8 | df-ioo 12179 | . . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
9 | df-ico 12181 | . . . . . 6 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | xrlenlt 10103 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐴)) | |
11 | xrlttr 11973 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑤 < 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞)) | |
12 | xrltletr 11988 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((-∞ < 𝐴 ∧ 𝐴 ≤ 𝑤) → -∞ < 𝑤)) | |
13 | 8, 9, 10, 8, 11, 12 | ixxun 12191 | . . . . 5 ⊢ (((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 𝐴 ∧ 𝐴 < +∞)) → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = (-∞(,)+∞)) |
14 | 2, 3, 5, 6, 7, 13 | syl32anc 1334 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = (-∞(,)+∞)) |
15 | ioomax 12248 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
16 | 14, 15 | syl6eq 2672 | . . 3 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ) |
17 | ioossre 12235 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ | |
18 | 8, 9, 10 | ixxdisj 12190 | . . . . 5 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅) |
19 | 2, 3, 5, 18 | syl3anc 1326 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅) |
20 | uneqdifeq 4057 | . . . 4 ⊢ (((-∞(,)𝐴) ⊆ ℝ ∧ ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅) → (((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞))) | |
21 | 17, 19, 20 | sylancr 695 | . . 3 ⊢ (𝐴 ∈ ℝ → (((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞))) |
22 | 16, 21 | mpbid 222 | . 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞)) |
23 | retop 22565 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
24 | iooretop 22569 | . . 3 ⊢ (-∞(,)𝐴) ∈ (topGen‘ran (,)) | |
25 | uniretop 22566 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
26 | 25 | opncld 20837 | . . 3 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)𝐴) ∈ (topGen‘ran (,))) → (ℝ ∖ (-∞(,)𝐴)) ∈ (Clsd‘(topGen‘ran (,)))) |
27 | 23, 24, 26 | mp2an 708 | . 2 ⊢ (ℝ ∖ (-∞(,)𝐴)) ∈ (Clsd‘(topGen‘ran (,))) |
28 | 22, 27 | syl6eqelr 2710 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 (,)cioo 12175 [,)cico 12177 topGenctg 16098 Topctop 20698 Clsdccld 20820 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-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-iun 4522 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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 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-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-ioo 12179 df-ico 12181 df-topgen 16104 df-top 20699 df-bases 20750 df-cld 20823 |
This theorem is referenced by: sxbrsigalem3 30334 orvcgteel 30529 dvasin 33496 dvacos 33497 dvreasin 33498 dvreacos 33499 rfcnpre3 39192 |
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