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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 10686 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ*) |
3 | rexr 10675 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 10683 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
6 | mnflt 12506 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
7 | ltpnf 12503 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
8 | df-ioc 12731 | . . . . . . 7 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
9 | df-ioo 12730 | . . . . . . 7 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | xrltnle 10696 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴)) | |
11 | xrlelttr 12537 | . . . . . . 7 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞)) | |
12 | xrlttr 12521 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((-∞ < 𝐴 ∧ 𝐴 < 𝑤) → -∞ < 𝑤)) | |
13 | 8, 9, 10, 9, 11, 12 | ixxun 12742 | . . . . . 6 ⊢ (((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 𝐴 ∧ 𝐴 < +∞)) → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
14 | 2, 3, 5, 6, 7, 13 | syl32anc 1370 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
15 | ioomax 12799 | . . . . 5 ⊢ (-∞(,)+∞) = ℝ | |
16 | 14, 15 | syl6eq 2869 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ) |
17 | iocssre 12804 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ) → (-∞(,]𝐴) ⊆ ℝ) | |
18 | 1, 17 | mpan 686 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ⊆ ℝ) |
19 | 8, 9, 10 | ixxdisj 12741 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
20 | 1, 3, 5, 19 | mp3an2i 1457 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
21 | uneqdifeq 4434 | . . . . 5 ⊢ (((-∞(,]𝐴) ⊆ ℝ ∧ ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) → (((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞))) | |
22 | 18, 20, 21 | syl2anc 584 | . . . 4 ⊢ (𝐴 ∈ ℝ → (((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞))) |
23 | 16, 22 | mpbid 233 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞)) |
24 | iooretop 23301 | . . 3 ⊢ (𝐴(,)+∞) ∈ (topGen‘ran (,)) | |
25 | 23, 24 | syl6eqel 2918 | . 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,))) |
26 | retop 23297 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
27 | uniretop 23298 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
28 | 27 | iscld2 21564 | . . 3 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,]𝐴) ⊆ ℝ) → ((-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,)))) |
29 | 26, 18, 28 | sylancr 587 | . 2 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,)))) |
30 | 25, 29 | mpbird 258 | 1 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 +∞cpnf 10660 -∞cmnf 10661 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 (,)cioo 12726 (,]cioc 12727 topGenctg 16699 Topctop 21429 Clsdccld 21552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-ioo 12730 df-ioc 12731 df-topgen 16705 df-top 21430 df-bases 21482 df-cld 21555 |
This theorem is referenced by: logdmopn 25159 orvclteel 31629 dvasin 34859 dvacos 34860 dvreasin 34861 dvreacos 34862 rfcnpre4 41168 |
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