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 2823 | . . . . . 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 41920 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵))) |
6 | eqid 2823 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 23394 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
9 | ax-resscn 10596 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
10 | unicntop 23396 | . . . . . . . 8 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
11 | 9, 10 | sseqtri 4005 | . . . . . . 7 ⊢ ℝ ⊆ ∪ (TopOpen‘ℂfld) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ∪ (TopOpen‘ℂfld)) |
13 | ioossre 12801 | . . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
15 | eqid 2823 | . . . . . . 7 ⊢ ∪ (TopOpen‘ℂfld) = ∪ (TopOpen‘ℂfld) | |
16 | 6 | tgioo2 23413 | . . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
17 | 15, 16 | restlp 21793 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → ((limPt‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
19 | 5, 18 | eleqtrd 2917 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ)) |
20 | elin 4171 | . . . 4 ⊢ (𝐴 ∈ (((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∩ ℝ) ↔ (𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐴 ∈ ℝ)) | |
21 | 19, 20 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) ∧ 𝐴 ∈ ℝ)) |
22 | 21 | simpld 497 | . 2 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵))) |
23 | lptioo1cn.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
24 | 23 | eqcomi 2832 | . . . 4 ⊢ (TopOpen‘ℂfld) = 𝐽 |
25 | 24 | fveq2i 6675 | . . 3 ⊢ (limPt‘(TopOpen‘ℂfld)) = (limPt‘𝐽) |
26 | 25 | fveq1i 6673 | . 2 ⊢ ((limPt‘(TopOpen‘ℂfld))‘(𝐴(,)𝐵)) = ((limPt‘𝐽)‘(𝐴(,)𝐵)) |
27 | 22, 26 | eleqtrdi 2925 | 1 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 ℝ*cxr 10676 < clt 10677 (,)cioo 12741 TopOpenctopn 16697 topGenctg 16713 ℂfldccnfld 20547 Topctop 21503 limPtclp 21744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-xms 22932 df-ms 22933 |
This theorem is referenced by: cncfiooiccre 42185 fourierdlem61 42459 fourierdlem75 42473 fourierdlem85 42483 fourierdlem88 42486 fourierdlem94 42492 fourierdlem95 42493 fourierdlem103 42501 fourierdlem104 42502 fourierdlem113 42511 |
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