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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lptioo1cn | Structured version Visualization version GIF version |
Description: The lower bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lptioo1cn.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
lptioo1cn.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
lptioo1cn.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lptioo1cn.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
lptioo1cn | ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . . . 6 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
2 | lptioo1cn.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | lptioo1cn.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
4 | lptioo1cn.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
5 | 1, 2, 3, 4 | lptioo1 45083 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) |
6 | eqid 2725 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 24718 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
9 | ax-resscn 11195 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
10 | unicntop 24720 | . . . . . . . 8 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
11 | 9, 10 | sseqtri 4009 | . . . . . . 7 ⊢ ℝ ⊆ ∪ (TopOpen‘ℂfld) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ∪ (TopOpen‘ℂfld)) |
13 | ioossre 13417 | . . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
15 | eqid 2725 | . . . . . . 7 ⊢ ∪ (TopOpen‘ℂfld) = ∪ (TopOpen‘ℂfld) | |
16 | 6 | tgioo2 24737 | . . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
17 | 15, 16 | restlp 23105 | . . . . . 6 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ ℝ ⊆ ∪ (TopOpen‘ℂfld) ∧ (𝐴(,)𝐵) ⊆ ℝ) → ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
18 | 8, 12, 14, 17 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
19 | 5, 18 | eleqtrd 2827 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
20 | elin 3955 | . . . 4 ⊢ (𝐴 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) ↔ (𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐴 ∈ ℝ)) | |
21 | 19, 20 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐴 ∈ ℝ)) |
22 | 21 | simpld 493 | . 2 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵))) |
23 | lptioo1cn.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
24 | 23 | eqcomi 2734 | . . . 4 ⊢ (TopOpen‘ℂfld) = 𝐽 |
25 | 24 | fveq2i 6895 | . . 3 ⊢ (limPt‘(TopOpen‘ℂfld)) = (limPt‘𝐽) |
26 | 25 | fveq1i 6893 | . 2 ⊢ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) = ((limPt‘𝐽)‘(𝐴(,)𝐵)) |
27 | 22, 26 | eleqtrdi 2835 | 1 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3938 ⊆ wss 3939 ∪ cuni 4903 class class class wbr 5143 ran crn 5673 ‘cfv 6543 (class class class)co 7416 ℂcc 11136 ℝcr 11137 ℝ*cxr 11277 < clt 11278 (,)cioo 13356 TopOpenctopn 17402 topGenctg 17418 ℂfldccnfld 21283 Topctop 22813 limPtclp 23056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fi 9434 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-rest 17403 df-topn 17404 df-topgen 17424 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-xms 24244 df-ms 24245 |
This theorem is referenced by: cncfiooiccre 45346 fourierdlem61 45618 fourierdlem75 45632 fourierdlem85 45642 fourierdlem88 45645 fourierdlem94 45651 fourierdlem95 45652 fourierdlem103 45660 fourierdlem104 45661 fourierdlem113 45670 |
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