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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lptioo2cn | Structured version Visualization version GIF version |
Description: The upper 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 |
---|---|
lptioo2cn.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
lptioo2cn.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
lptioo2cn.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lptioo2cn.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
lptioo2cn | ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . . . . 6 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
2 | lptioo2cn.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
3 | lptioo2cn.3 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | lptioo2cn.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
5 | 1, 2, 3, 4 | lptioo2 40607 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) |
6 | eqid 2799 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 22915 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | ax-resscn 10281 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
9 | unicntop 22917 | . . . . . . 7 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
10 | 8, 9 | sseqtri 3833 | . . . . . 6 ⊢ ℝ ⊆ ∪ (TopOpen‘ℂfld) |
11 | ioossre 12484 | . . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
12 | eqid 2799 | . . . . . . 7 ⊢ ∪ (TopOpen‘ℂfld) = ∪ (TopOpen‘ℂfld) | |
13 | 6 | tgioo2 22934 | . . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
14 | 12, 13 | restlp 21316 | . . . . . 6 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ ℝ ⊆ ∪ (TopOpen‘ℂfld) ∧ (𝐴(,)𝐵) ⊆ ℝ) → ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
15 | 7, 10, 11, 14 | mp3an 1586 | . . . . 5 ⊢ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) |
16 | 5, 15 | syl6eleq 2888 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
17 | elin 3994 | . . . 4 ⊢ (𝐵 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) ↔ (𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐵 ∈ ℝ)) | |
18 | 16, 17 | sylib 210 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐵 ∈ ℝ)) |
19 | 18 | simpld 489 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵))) |
20 | lptioo2cn.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
21 | 20 | eqcomi 2808 | . . . 4 ⊢ (TopOpen‘ℂfld) = 𝐽 |
22 | 21 | fveq2i 6414 | . . 3 ⊢ (limPt‘(TopOpen‘ℂfld)) = (limPt‘𝐽) |
23 | 22 | fveq1i 6412 | . 2 ⊢ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) = ((limPt‘𝐽)‘(𝐴(,)𝐵)) |
24 | 19, 23 | syl6eleq 2888 | 1 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∩ cin 3768 ⊆ wss 3769 ∪ cuni 4628 class class class wbr 4843 ran crn 5313 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 ℝ*cxr 10362 < clt 10363 (,)cioo 12424 TopOpenctopn 16397 topGenctg 16413 ℂfldccnfld 20068 Topctop 21026 limPtclp 21267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fi 8559 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-fz 12581 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-plusg 16280 df-mulr 16281 df-starv 16282 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-rest 16398 df-topn 16399 df-topgen 16419 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cld 21152 df-ntr 21153 df-cls 21154 df-nei 21231 df-lp 21269 df-xms 22453 df-ms 22454 |
This theorem is referenced by: cncfiooiccre 40852 fourierdlem60 41126 fourierdlem74 41140 fourierdlem88 41154 fourierdlem94 41160 fourierdlem95 41161 fourierdlem103 41169 fourierdlem104 41170 fourierdlem113 41179 |
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