Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > icccld | Structured version Visualization version GIF version |
Description: Closed intervals are closed sets of the standard topology on ℝ. (Contributed by FL, 14-Sep-2007.) |
Ref | Expression |
---|---|
icccld | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difreicc 13317 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) = ((-∞(,)𝐴) ∪ (𝐵(,)+∞))) | |
2 | retop 24031 | . . . 4 ⊢ (topGen‘ran (,)) ∈ Top | |
3 | iooretop 24035 | . . . 4 ⊢ (-∞(,)𝐴) ∈ (topGen‘ran (,)) | |
4 | iooretop 24035 | . . . 4 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,)) | |
5 | unopn 22158 | . . . 4 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)𝐴) ∈ (topGen‘ran (,)) ∧ (𝐵(,)+∞) ∈ (topGen‘ran (,))) → ((-∞(,)𝐴) ∪ (𝐵(,)+∞)) ∈ (topGen‘ran (,))) | |
6 | 2, 3, 4, 5 | mp3an 1460 | . . 3 ⊢ ((-∞(,)𝐴) ∪ (𝐵(,)+∞)) ∈ (topGen‘ran (,)) |
7 | 1, 6 | eqeltrdi 2845 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) ∈ (topGen‘ran (,))) |
8 | iccssre 13262 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
9 | uniretop 24032 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
10 | 9 | iscld2 22285 | . . 3 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → ((𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (𝐴[,]𝐵)) ∈ (topGen‘ran (,)))) |
11 | 2, 8, 10 | sylancr 587 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))) ↔ (ℝ ∖ (𝐴[,]𝐵)) ∈ (topGen‘ran (,)))) |
12 | 7, 11 | mpbird 256 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2105 ∖ cdif 3895 ∪ cun 3896 ⊆ wss 3898 ran crn 5621 ‘cfv 6479 (class class class)co 7337 ℝcr 10971 +∞cpnf 11107 -∞cmnf 11108 (,)cioo 13180 [,]cicc 13183 topGenctg 17245 Topctop 22148 Clsdccld 22273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-inf 9300 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-n0 12335 df-z 12421 df-uz 12684 df-q 12790 df-ioo 13184 df-icc 13187 df-topgen 17251 df-top 22149 df-bases 22202 df-cld 22276 |
This theorem is referenced by: cnmpopc 24197 cvmliftlem10 33555 mblfinlem1 35927 mblfinlem2 35928 icccmpALT 36112 |
Copyright terms: Public domain | W3C validator |