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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 11308 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ*) |
3 | rexr 11297 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 11305 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
6 | mnflt 13143 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
7 | ltpnf 13140 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
8 | df-ioc 13369 | . . . . . . 7 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
9 | df-ioo 13368 | . . . . . . 7 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | xrltnle 11318 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴)) | |
11 | xrlelttr 13175 | . . . . . . 7 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞)) | |
12 | xrlttr 13159 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((-∞ < 𝐴 ∧ 𝐴 < 𝑤) → -∞ < 𝑤)) | |
13 | 8, 9, 10, 9, 11, 12 | ixxun 13380 | . . . . . 6 ⊢ (((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 𝐴 ∧ 𝐴 < +∞)) → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
14 | 2, 3, 5, 6, 7, 13 | syl32anc 1375 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
15 | ioomax 13439 | . . . . 5 ⊢ (-∞(,)+∞) = ℝ | |
16 | 14, 15 | eqtrdi 2781 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ) |
17 | iocssre 13444 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ) → (-∞(,]𝐴) ⊆ ℝ) | |
18 | 1, 17 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ⊆ ℝ) |
19 | 8, 9, 10 | ixxdisj 13379 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
20 | 1, 3, 5, 19 | mp3an2i 1462 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
21 | uneqdifeq 4494 | . . . . 5 ⊢ (((-∞(,]𝐴) ⊆ ℝ ∧ ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) → (((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞))) | |
22 | 18, 20, 21 | syl2anc 582 | . . . 4 ⊢ (𝐴 ∈ ℝ → (((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞))) |
23 | 16, 22 | mpbid 231 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞)) |
24 | iooretop 24743 | . . 3 ⊢ (𝐴(,)+∞) ∈ (topGen‘ran (,)) | |
25 | 23, 24 | eqeltrdi 2833 | . 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,))) |
26 | retop 24739 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
27 | uniretop 24740 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
28 | 27 | iscld2 22993 | . . 3 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,]𝐴) ⊆ ℝ) → ((-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,)))) |
29 | 26, 18, 28 | sylancr 585 | . 2 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,)))) |
30 | 25, 29 | mpbird 256 | 1 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 class class class wbr 5149 ran crn 5679 ‘cfv 6549 (class class class)co 7419 ℝcr 11144 +∞cpnf 11282 -∞cmnf 11283 ℝ*cxr 11284 < clt 11285 ≤ cle 11286 (,)cioo 13364 (,]cioc 13365 topGenctg 17438 Topctop 22856 Clsdccld 22981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9472 df-inf 9473 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-ioo 13368 df-ioc 13369 df-topgen 17444 df-top 22857 df-bases 22910 df-cld 22984 |
This theorem is referenced by: logdmopn 26645 orvclteel 34243 dvasin 37328 dvacos 37329 dvreasin 37330 dvreacos 37331 rfcnpre4 44543 |
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