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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 2778 | . . . . . 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 40772 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) |
6 | eqid 2778 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 22995 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
9 | ax-resscn 10329 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
10 | unicntop 22997 | . . . . . . . 8 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
11 | 9, 10 | sseqtri 3856 | . . . . . . 7 ⊢ ℝ ⊆ ∪ (TopOpen‘ℂfld) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ∪ (TopOpen‘ℂfld)) |
13 | ioossre 12547 | . . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
15 | eqid 2778 | . . . . . . 7 ⊢ ∪ (TopOpen‘ℂfld) = ∪ (TopOpen‘ℂfld) | |
16 | 6 | tgioo2 23014 | . . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
17 | 15, 16 | restlp 21395 | . . . . . 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 1439 | . . . . 5 ⊢ (𝜑 → ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
19 | 5, 18 | eleqtrd 2861 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
20 | elin 4019 | . . . 4 ⊢ (𝐴 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) ↔ (𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐴 ∈ ℝ)) | |
21 | 19, 20 | sylib 210 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐴 ∈ ℝ)) |
22 | 21 | simpld 490 | . 2 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵))) |
23 | lptioo1cn.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
24 | 23 | eqcomi 2787 | . . . 4 ⊢ (TopOpen‘ℂfld) = 𝐽 |
25 | 24 | fveq2i 6449 | . . 3 ⊢ (limPt‘(TopOpen‘ℂfld)) = (limPt‘𝐽) |
26 | 25 | fveq1i 6447 | . 2 ⊢ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) = ((limPt‘𝐽)‘(𝐴(,)𝐵)) |
27 | 22, 26 | syl6eleq 2869 | 1 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ⊆ wss 3792 ∪ cuni 4671 class class class wbr 4886 ran crn 5356 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 ℝ*cxr 10410 < clt 10411 (,)cioo 12487 TopOpenctopn 16468 topGenctg 16484 ℂfldccnfld 20142 Topctop 21105 limPtclp 21346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-topn 16470 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-xms 22533 df-ms 22534 |
This theorem is referenced by: cncfiooiccre 41036 fourierdlem61 41311 fourierdlem75 41325 fourierdlem85 41335 fourierdlem88 41338 fourierdlem94 41344 fourierdlem95 41345 fourierdlem103 41353 fourierdlem104 41354 fourierdlem113 41363 |
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