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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 40782 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) |
6 | eqid 2778 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 22999 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
9 | ax-resscn 10331 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
10 | unicntop 23001 | . . . . . . . 8 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
11 | 9, 10 | sseqtri 3856 | . . . . . . 7 ⊢ ℝ ⊆ ∪ (TopOpen‘ℂfld) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ∪ (TopOpen‘ℂfld)) |
13 | ioossre 12551 | . . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
15 | eqid 2778 | . . . . . . 7 ⊢ ∪ (TopOpen‘ℂfld) = ∪ (TopOpen‘ℂfld) | |
16 | 6 | tgioo2 23018 | . . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
17 | 15, 16 | restlp 21399 | . . . . . 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 6451 | . . 3 ⊢ (limPt‘(TopOpen‘ℂfld)) = (limPt‘𝐽) |
26 | 25 | fveq1i 6449 | . 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 4673 class class class wbr 4888 ran crn 5358 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 ℝ*cxr 10412 < clt 10413 (,)cioo 12491 TopOpenctopn 16472 topGenctg 16488 ℂfldccnfld 20146 Topctop 21109 limPtclp 21350 |
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 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
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 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fi 8607 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-fz 12648 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-plusg 16355 df-mulr 16356 df-starv 16357 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-rest 16473 df-topn 16474 df-topgen 16494 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-lp 21352 df-xms 22537 df-ms 22538 |
This theorem is referenced by: cncfiooiccre 41046 fourierdlem61 41321 fourierdlem75 41335 fourierdlem85 41345 fourierdlem88 41348 fourierdlem94 41354 fourierdlem95 41355 fourierdlem103 41363 fourierdlem104 41364 fourierdlem113 41373 |
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