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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ioodvbdlimc2 | Structured version Visualization version GIF version |
Description: A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.) |
Ref | Expression |
---|---|
ioodvbdlimc2.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ioodvbdlimc2.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ioodvbdlimc2.f | ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) |
ioodvbdlimc2.dmdv | ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) |
ioodvbdlimc2.dvbd | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦) |
Ref | Expression |
---|---|
ioodvbdlimc2 | ⊢ (𝜑 → (𝐹 limℂ 𝐵) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioodvbdlimc2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ) |
3 | ioodvbdlimc2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ) |
5 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
6 | ioodvbdlimc2.f | . . . . 5 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) | |
7 | 6 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐹:(𝐴(,)𝐵)⟶ℝ) |
8 | ioodvbdlimc2.dmdv | . . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) | |
9 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) |
10 | ioodvbdlimc2.dvbd | . . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦) | |
11 | 10 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦) |
12 | 2fveq3 6650 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (abs‘((ℝ D 𝐹)‘𝑦)) = (abs‘((ℝ D 𝐹)‘𝑥))) | |
13 | 12 | cbvmptv 5133 | . . . . . 6 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))) |
14 | 13 | rneqi 5771 | . . . . 5 ⊢ ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))) = ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))) |
15 | 14 | supeq1i 8895 | . . . 4 ⊢ sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) |
16 | eqid 2798 | . . . 4 ⊢ ((⌊‘(1 / (𝐵 − 𝐴))) + 1) = ((⌊‘(1 / (𝐵 − 𝐴))) + 1) | |
17 | oveq2 7143 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (1 / 𝑘) = (1 / 𝑗)) | |
18 | 17 | oveq2d 7151 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 − (1 / 𝑘)) = (𝐵 − (1 / 𝑗))) |
19 | 18 | fveq2d 6649 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝐵 − (1 / 𝑘))) = (𝐹‘(𝐵 − (1 / 𝑗)))) |
20 | 19 | cbvmptv 5133 | . . . 4 ⊢ (𝑘 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘)))) = (𝑗 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑗)))) |
21 | 18 | cbvmptv 5133 | . . . 4 ⊢ (𝑘 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐵 − (1 / 𝑘))) = (𝑗 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐵 − (1 / 𝑗))) |
22 | eqid 2798 | . . . 4 ⊢ if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1)) = if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1)) | |
23 | biid 264 | . . . 4 ⊢ (((((((𝜑 ∧ 𝐴 < 𝐵) ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ (ℤ≥‘if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1)))) ∧ (abs‘(((𝑘 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))‘𝑗) − (lim sup‘(𝑘 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐵)) < (1 / 𝑗)) ↔ ((((((𝜑 ∧ 𝐴 < 𝐵) ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ (ℤ≥‘if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1)))) ∧ (abs‘(((𝑘 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))‘𝑗) − (lim sup‘(𝑘 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐵)) < (1 / 𝑗))) | |
24 | 2, 4, 5, 7, 9, 11, 15, 16, 20, 21, 22, 23 | ioodvbdlimc2lem 42576 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (lim sup‘(𝑘 ∈ (ℤ≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))) ∈ (𝐹 limℂ 𝐵)) |
25 | 24 | ne0d 4251 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (𝐹 limℂ 𝐵) ≠ ∅) |
26 | ax-resscn 10583 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
27 | 26 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
28 | 6, 27 | fssd 6502 | . . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ) |
29 | 28 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐹:(𝐴(,)𝐵)⟶ℂ) |
30 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴) | |
31 | 1 | rexrd 10680 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
32 | 31 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ*) |
33 | 3 | rexrd 10680 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
34 | 33 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ*) |
35 | ioo0 12751 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | |
36 | 32, 34, 35 | syl2anc 587 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) |
37 | 30, 36 | mpbird 260 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐴(,)𝐵) = ∅) |
38 | 37 | feq2d 6473 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐹:(𝐴(,)𝐵)⟶ℂ ↔ 𝐹:∅⟶ℂ)) |
39 | 29, 38 | mpbid 235 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐹:∅⟶ℂ) |
40 | 3 | recnd 10658 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
41 | 40 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℂ) |
42 | 39, 41 | limcdm0 42260 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐹 limℂ 𝐵) = ℂ) |
43 | 0cn 10622 | . . . . 5 ⊢ 0 ∈ ℂ | |
44 | 43 | ne0ii 4253 | . . . 4 ⊢ ℂ ≠ ∅ |
45 | 44 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ℂ ≠ ∅) |
46 | 42, 45 | eqnetrd 3054 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐹 limℂ 𝐵) ≠ ∅) |
47 | 25, 46, 1, 3 | ltlecasei 10737 | 1 ⊢ (𝜑 → (𝐹 limℂ 𝐵) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ∅c0 4243 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ℤ≥cuz 12231 ℝ+crp 12377 (,)cioo 12726 ⌊cfl 13155 abscabs 14585 lim supclsp 14819 limℂ climc 24465 D cdv 24466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 |
This theorem is referenced by: fourierdlem94 42842 fourierdlem113 42861 |
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