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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 2734 | . . . . . 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 45486 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) |
6 | eqid 2734 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 24818 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | ax-resscn 11237 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
9 | unicntop 24820 | . . . . . . 7 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
10 | 8, 9 | sseqtri 4039 | . . . . . 6 ⊢ ℝ ⊆ ∪ (TopOpen‘ℂfld) |
11 | ioossre 13464 | . . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
12 | eqid 2734 | . . . . . . 7 ⊢ ∪ (TopOpen‘ℂfld) = ∪ (TopOpen‘ℂfld) | |
13 | 6 | tgioo2 24837 | . . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
14 | 12, 13 | restlp 23205 | . . . . . 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 1461 | . . . . 5 ⊢ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) |
16 | 5, 15 | eleqtrdi 2848 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
17 | elin 3986 | . . . 4 ⊢ (𝐵 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) ↔ (𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐵 ∈ ℝ)) | |
18 | 16, 17 | sylib 218 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐵 ∈ ℝ)) |
19 | 18 | simpld 494 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵))) |
20 | lptioo2cn.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
21 | 20 | eqcomi 2743 | . . . 4 ⊢ (TopOpen‘ℂfld) = 𝐽 |
22 | 21 | fveq2i 6922 | . . 3 ⊢ (limPt‘(TopOpen‘ℂfld)) = (limPt‘𝐽) |
23 | 22 | fveq1i 6920 | . 2 ⊢ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) = ((limPt‘𝐽)‘(𝐴(,)𝐵)) |
24 | 19, 23 | eleqtrdi 2848 | 1 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∩ cin 3969 ⊆ wss 3970 ∪ cuni 4931 class class class wbr 5169 ran crn 5700 ‘cfv 6572 (class class class)co 7445 ℂcc 11178 ℝcr 11179 ℝ*cxr 11319 < clt 11320 (,)cioo 13403 TopOpenctopn 17476 topGenctg 17492 ℂfldccnfld 21382 Topctop 22913 limPtclp 23156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fi 9476 df-sup 9507 df-inf 9508 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-fz 13564 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-struct 17189 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-mulr 17320 df-starv 17321 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-rest 17477 df-topn 17478 df-topgen 17498 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-lp 23158 df-xms 24344 df-ms 24345 |
This theorem is referenced by: cncfiooiccre 45750 fourierdlem60 46021 fourierdlem74 46035 fourierdlem88 46049 fourierdlem94 46055 fourierdlem95 46056 fourierdlem103 46064 fourierdlem104 46065 fourierdlem113 46074 |
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