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Mirrors > Home > MPE Home > Th. List > icopnfcld | Structured version Visualization version GIF version |
Description: Right-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
icopnfcld | ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 9975 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ*) |
3 | rexr 9964 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
4 | pnfxr 9971 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
6 | mnflt 11833 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
7 | ltpnf 11830 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
8 | df-ioo 12050 | . . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
9 | df-ico 12052 | . . . . . 6 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | xrlenlt 9982 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐴)) | |
11 | xrlttr 11849 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑤 < 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞)) | |
12 | xrltletr 11864 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((-∞ < 𝐴 ∧ 𝐴 ≤ 𝑤) → -∞ < 𝑤)) | |
13 | 8, 9, 10, 8, 11, 12 | ixxun 12062 | . . . . 5 ⊢ (((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 𝐴 ∧ 𝐴 < +∞)) → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = (-∞(,)+∞)) |
14 | 2, 3, 5, 6, 7, 13 | syl32anc 1326 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = (-∞(,)+∞)) |
15 | ioomax 12119 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
16 | 14, 15 | syl6eq 2660 | . . 3 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ) |
17 | ioossre 12106 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ | |
18 | 8, 9, 10 | ixxdisj 12061 | . . . . 5 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅) |
19 | 2, 3, 5, 18 | syl3anc 1318 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅) |
20 | uneqdifeq 4009 | . . . 4 ⊢ (((-∞(,)𝐴) ⊆ ℝ ∧ ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅) → (((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞))) | |
21 | 17, 19, 20 | sylancr 694 | . . 3 ⊢ (𝐴 ∈ ℝ → (((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞))) |
22 | 16, 21 | mpbid 221 | . 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞)) |
23 | retop 22375 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
24 | iooretop 22379 | . . 3 ⊢ (-∞(,)𝐴) ∈ (topGen‘ran (,)) | |
25 | uniretop 22376 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
26 | 25 | opncld 20647 | . . 3 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)𝐴) ∈ (topGen‘ran (,))) → (ℝ ∖ (-∞(,)𝐴)) ∈ (Clsd‘(topGen‘ran (,)))) |
27 | 23, 24, 26 | mp2an 704 | . 2 ⊢ (ℝ ∖ (-∞(,)𝐴)) ∈ (Clsd‘(topGen‘ran (,))) |
28 | 22, 27 | syl6eqelr 2697 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 [,)cico 12048 topGenctg 15921 Topctop 20517 Clsdccld 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050 df-ico 12052 df-topgen 15927 df-top 20521 df-bases 20522 df-cld 20633 |
This theorem is referenced by: sxbrsigalem3 29661 orvcgteel 29856 dvasin 32666 dvacos 32667 dvreasin 32668 dvreacos 32669 rfcnpre3 38215 |
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