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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 11033 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ*) |
3 | rexr 11022 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 11030 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
6 | mnflt 12858 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
7 | ltpnf 12855 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
8 | df-ioc 13083 | . . . . . . 7 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
9 | df-ioo 13082 | . . . . . . 7 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | xrltnle 11043 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴)) | |
11 | xrlelttr 12889 | . . . . . . 7 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞)) | |
12 | xrlttr 12873 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((-∞ < 𝐴 ∧ 𝐴 < 𝑤) → -∞ < 𝑤)) | |
13 | 8, 9, 10, 9, 11, 12 | ixxun 13094 | . . . . . 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 13153 | . . . . 5 ⊢ (-∞(,)+∞) = ℝ | |
16 | 14, 15 | eqtrdi 2796 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∪ (𝐴(,)+∞)) = ℝ) |
17 | iocssre 13158 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ) → (-∞(,]𝐴) ⊆ ℝ) | |
18 | 1, 17 | mpan 687 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ⊆ ℝ) |
19 | 8, 9, 10 | ixxdisj 13093 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
20 | 1, 3, 5, 19 | mp3an2i 1465 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((-∞(,]𝐴) ∩ (𝐴(,)+∞)) = ∅) |
21 | uneqdifeq 4429 | . . . . 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 231 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) = (𝐴(,)+∞)) |
24 | iooretop 23927 | . . 3 ⊢ (𝐴(,)+∞) ∈ (topGen‘ran (,)) | |
25 | 23, 24 | eqeltrdi 2849 | . 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,]𝐴)) ∈ (topGen‘ran (,))) |
26 | retop 23923 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
27 | uniretop 23924 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
28 | 27 | iscld2 22177 | . . 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 256 | 1 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2110 ∖ cdif 3889 ∪ cun 3890 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 class class class wbr 5079 ran crn 5591 ‘cfv 6432 (class class class)co 7271 ℝcr 10871 +∞cpnf 11007 -∞cmnf 11008 ℝ*cxr 11009 < clt 11010 ≤ cle 11011 (,)cioo 13078 (,]cioc 13079 topGenctg 17146 Topctop 22040 Clsdccld 22165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12582 df-q 12688 df-ioo 13082 df-ioc 13083 df-topgen 17152 df-top 22041 df-bases 22094 df-cld 22168 |
This theorem is referenced by: logdmopn 25802 orvclteel 32435 dvasin 35857 dvacos 35858 dvreasin 35859 dvreacos 35860 rfcnpre4 42547 |
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