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Mirrors > Home > MPE Home > Th. List > iocmnfcld | Structured version Visualization version GIF version |
Description: Left-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
iocmnfcld | ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 10386 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ*) |
3 | rexr 10374 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 10382 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
6 | mnflt 12204 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
7 | ltpnf 12201 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
8 | df-ioc 12429 | . . . . . . 7 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
9 | df-ioo 12428 | . . . . . . 7 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | xrltnle 10395 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴)) | |
11 | xrlelttr 12236 | . . . . . . 7 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞)) | |
12 | xrlttr 12220 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((-∞ < 𝐴 ∧ 𝐴 < 𝑤) → -∞ < 𝑤)) | |
13 | 8, 9, 10, 9, 11, 12 | ixxun 12440 | . . . . . 6 ⊢ (((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 𝐴 ∧ 𝐴 < +∞)) → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
14 | 2, 3, 5, 6, 7, 13 | syl32anc 1498 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
15 | ioomax 12497 | . . . . 5 ⊢ (-∞(,)+∞) = ℝ | |
16 | 14, 15 | syl6eq 2849 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ) |
17 | iocssre 12502 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ) → (-∞(,]𝐴) ⊆ ℝ) | |
18 | 1, 17 | mpan 682 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ⊆ ℝ) |
19 | 8, 9, 10 | ixxdisj 12439 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
20 | 2, 3, 5, 19 | syl3anc 1491 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
21 | uneqdifeq 4251 | . . . . 5 ⊢ (((-∞(,]𝐴) ⊆ ℝ ∧ ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) → (((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞))) | |
22 | 18, 20, 21 | syl2anc 580 | . . . 4 ⊢ (𝐴 ∈ ℝ → (((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞))) |
23 | 16, 22 | mpbid 224 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞)) |
24 | iooretop 22897 | . . 3 ⊢ (𝐴(,)+∞) ∈ (topGen‘ran (,)) | |
25 | 23, 24 | syl6eqel 2886 | . 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,))) |
26 | retop 22893 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
27 | uniretop 22894 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
28 | 27 | iscld2 21161 | . . 3 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,]𝐴) ⊆ ℝ) → ((-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,)))) |
29 | 26, 18, 28 | sylancr 582 | . 2 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,)))) |
30 | 25, 29 | mpbird 249 | 1 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ∖ cdif 3766 ∪ cun 3767 ∩ cin 3768 ⊆ wss 3769 ∅c0 4115 class class class wbr 4843 ran crn 5313 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 +∞cpnf 10360 -∞cmnf 10361 ℝ*cxr 10362 < clt 10363 ≤ cle 10364 (,)cioo 12424 (,]cioc 12425 topGenctg 16413 Topctop 21026 Clsdccld 21149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-ioo 12428 df-ioc 12429 df-topgen 16419 df-top 21027 df-bases 21079 df-cld 21152 |
This theorem is referenced by: logdmopn 24736 orvclteel 31051 dvasin 33984 dvacos 33985 dvreasin 33986 dvreacos 33987 rfcnpre4 39953 |
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