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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 11315 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ*) |
3 | rexr 11304 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 11312 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
6 | mnflt 13162 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
7 | ltpnf 13159 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
8 | df-ioc 13388 | . . . . . . 7 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
9 | df-ioo 13387 | . . . . . . 7 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | xrltnle 11325 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴)) | |
11 | xrlelttr 13194 | . . . . . . 7 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞)) | |
12 | xrlttr 13178 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((-∞ < 𝐴 ∧ 𝐴 < 𝑤) → -∞ < 𝑤)) | |
13 | 8, 9, 10, 9, 11, 12 | ixxun 13399 | . . . . . 6 ⊢ (((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 𝐴 ∧ 𝐴 < +∞)) → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
14 | 2, 3, 5, 6, 7, 13 | syl32anc 1377 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = (-∞(,)+∞)) |
15 | ioomax 13458 | . . . . 5 ⊢ (-∞(,)+∞) = ℝ | |
16 | 14, 15 | eqtrdi 2790 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ) |
17 | iocssre 13463 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ) → (-∞(,]𝐴) ⊆ ℝ) | |
18 | 1, 17 | mpan 690 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ⊆ ℝ) |
19 | 8, 9, 10 | ixxdisj 13398 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
20 | 1, 3, 5, 19 | mp3an2i 1465 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
21 | uneqdifeq 4498 | . . . . 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 232 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞)) |
24 | iooretop 24801 | . . 3 ⊢ (𝐴(,)+∞) ∈ (topGen‘ran (,)) | |
25 | 23, 24 | eqeltrdi 2846 | . 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,))) |
26 | retop 24797 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
27 | uniretop 24798 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
28 | 27 | iscld2 23051 | . . 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 257 | 1 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 +∞cpnf 11289 -∞cmnf 11290 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 (,)cioo 13383 (,]cioc 13384 topGenctg 17483 Topctop 22914 Clsdccld 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-ioo 13387 df-ioc 13388 df-topgen 17489 df-top 22915 df-bases 22968 df-cld 23042 |
This theorem is referenced by: logdmopn 26705 orvclteel 34453 dvasin 37690 dvacos 37691 dvreasin 37692 dvreacos 37693 rfcnpre4 44971 |
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