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 2738 | . . . . . 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 42714 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) |
6 | eqid 2738 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 23536 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | ax-resscn 10672 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
9 | unicntop 23538 | . . . . . . 7 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
10 | 8, 9 | sseqtri 3913 | . . . . . 6 ⊢ ℝ ⊆ ∪ (TopOpen‘ℂfld) |
11 | ioossre 12882 | . . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
12 | eqid 2738 | . . . . . . 7 ⊢ ∪ (TopOpen‘ℂfld) = ∪ (TopOpen‘ℂfld) | |
13 | 6 | tgioo2 23555 | . . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
14 | 12, 13 | restlp 21934 | . . . . . 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 1462 | . . . . 5 ⊢ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) |
16 | 5, 15 | eleqtrdi 2843 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
17 | elin 3859 | . . . 4 ⊢ (𝐵 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) ↔ (𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐵 ∈ ℝ)) | |
18 | 16, 17 | sylib 221 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐵 ∈ ℝ)) |
19 | 18 | simpld 498 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵))) |
20 | lptioo2cn.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
21 | 20 | eqcomi 2747 | . . . 4 ⊢ (TopOpen‘ℂfld) = 𝐽 |
22 | 21 | fveq2i 6677 | . . 3 ⊢ (limPt‘(TopOpen‘ℂfld)) = (limPt‘𝐽) |
23 | 22 | fveq1i 6675 | . 2 ⊢ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) = ((limPt‘𝐽)‘(𝐴(,)𝐵)) |
24 | 19, 23 | eleqtrdi 2843 | 1 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 ⊆ wss 3843 ∪ cuni 4796 class class class wbr 5030 ran crn 5526 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 ℝcr 10614 ℝ*cxr 10752 < clt 10753 (,)cioo 12821 TopOpenctopn 16798 topGenctg 16814 ℂfldccnfld 20217 Topctop 21644 limPtclp 21885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fi 8948 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-fz 12982 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-plusg 16681 df-mulr 16682 df-starv 16683 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-rest 16799 df-topn 16800 df-topgen 16820 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-cnfld 20218 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-lp 21887 df-xms 23073 df-ms 23074 |
This theorem is referenced by: cncfiooiccre 42978 fourierdlem60 43249 fourierdlem74 43263 fourierdlem88 43277 fourierdlem94 43283 fourierdlem95 43284 fourierdlem103 43292 fourierdlem104 43293 fourierdlem113 43302 |
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