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Mirrors > Home > MPE Home > Th. List > c1lip2 | Structured version Visualization version GIF version |
Description: C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
c1lip2.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
c1lip2.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
c1lip2.f | ⊢ (𝜑 → 𝐹 ∈ ((Cn‘ℝ)‘1)) |
c1lip2.rn | ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
c1lip2.dm | ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) |
Ref | Expression |
---|---|
c1lip2 | ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c1lip2.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | c1lip2.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | c1lip2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((Cn‘ℝ)‘1)) | |
4 | ax-resscn 10447 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
5 | 1nn0 11767 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
6 | elcpn 24218 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℕ0) → (𝐹 ∈ ((Cn‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ)))) | |
7 | 4, 5, 6 | mp2an 688 | . . . 4 ⊢ (𝐹 ∈ ((Cn‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ))) |
8 | 7 | simplbi 498 | . . 3 ⊢ (𝐹 ∈ ((Cn‘ℝ)‘1) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
9 | 3, 8 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ)) |
10 | c1lip2.dm | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) | |
11 | pmfun 8283 | . . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → Fun 𝐹) | |
12 | 9, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
13 | 12 | funfnd 6263 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
14 | c1lip2.rn | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) | |
15 | df-f 6236 | . . . . . . 7 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) | |
16 | 13, 14, 15 | sylanbrc 583 | . . . . . 6 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
17 | cnex 10471 | . . . . . . . . 9 ⊢ ℂ ∈ V | |
18 | reex 10481 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
19 | 17, 18 | elpm2 8295 | . . . . . . . 8 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
20 | 19 | simprbi 497 | . . . . . . 7 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ) |
21 | 9, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ) |
22 | dvfre 24235 | . . . . . 6 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) | |
23 | 16, 21, 22 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
24 | 0p1e1 11613 | . . . . . . . . . . 11 ⊢ (0 + 1) = 1 | |
25 | 24 | fveq2i 6548 | . . . . . . . . . 10 ⊢ ((ℝ D𝑛 𝐹)‘(0 + 1)) = ((ℝ D𝑛 𝐹)‘1) |
26 | 0nn0 11766 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0 | |
27 | dvnp1 24209 | . . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℝ) ∧ 0 ∈ ℕ0) → ((ℝ D𝑛 𝐹)‘(0 + 1)) = (ℝ D ((ℝ D𝑛 𝐹)‘0))) | |
28 | 4, 26, 27 | mp3an13 1444 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → ((ℝ D𝑛 𝐹)‘(0 + 1)) = (ℝ D ((ℝ D𝑛 𝐹)‘0))) |
29 | 9, 28 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘(0 + 1)) = (ℝ D ((ℝ D𝑛 𝐹)‘0))) |
30 | 25, 29 | syl5eqr 2847 | . . . . . . . . 9 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘1) = (ℝ D ((ℝ D𝑛 𝐹)‘0))) |
31 | dvn0 24208 | . . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℝ)) → ((ℝ D𝑛 𝐹)‘0) = 𝐹) | |
32 | 4, 9, 31 | sylancr 587 | . . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘0) = 𝐹) |
33 | 32 | oveq2d 7039 | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D ((ℝ D𝑛 𝐹)‘0)) = (ℝ D 𝐹)) |
34 | 30, 33 | eqtrd 2833 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘1) = (ℝ D 𝐹)) |
35 | 7 | simprbi 497 | . . . . . . . . 9 ⊢ (𝐹 ∈ ((Cn‘ℝ)‘1) → ((ℝ D𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ)) |
36 | 3, 35 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ)) |
37 | 34, 36 | eqeltrrd 2886 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ)) |
38 | cncff 23188 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ) | |
39 | fdm 6397 | . . . . . . 7 ⊢ ((ℝ D 𝐹):dom 𝐹⟶ℂ → dom (ℝ D 𝐹) = dom 𝐹) | |
40 | 37, 38, 39 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐹) |
41 | 40 | feq2d 6375 | . . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom 𝐹⟶ℝ)) |
42 | 23, 41 | mpbid 233 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom 𝐹⟶ℝ) |
43 | cncffvrn 23193 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ)) → ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ) ↔ (ℝ D 𝐹):dom 𝐹⟶ℝ)) | |
44 | 4, 37, 43 | sylancr 587 | . . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ) ↔ (ℝ D 𝐹):dom 𝐹⟶ℝ)) |
45 | 42, 44 | mpbird 258 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ)) |
46 | rescncf 23192 | . . 3 ⊢ ((𝐴[,]𝐵) ⊆ dom 𝐹 → ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))) | |
47 | 10, 45, 46 | sylc 65 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
48 | 18 | prid1 4611 | . . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ} |
49 | 1eluzge0 12145 | . . . . . . . . 9 ⊢ 1 ∈ (ℤ≥‘0) | |
50 | cpnord 24219 | . . . . . . . . 9 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ0 ∧ 1 ∈ (ℤ≥‘0)) → ((Cn‘ℝ)‘1) ⊆ ((Cn‘ℝ)‘0)) | |
51 | 48, 26, 49, 50 | mp3an 1453 | . . . . . . . 8 ⊢ ((Cn‘ℝ)‘1) ⊆ ((Cn‘ℝ)‘0) |
52 | 51, 3 | sseldi 3893 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((Cn‘ℝ)‘0)) |
53 | elcpn 24218 | . . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ 0 ∈ ℕ0) → (𝐹 ∈ ((Cn‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)))) | |
54 | 4, 26, 53 | mp2an 688 | . . . . . . . 8 ⊢ (𝐹 ∈ ((Cn‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))) |
55 | 54 | simprbi 497 | . . . . . . 7 ⊢ (𝐹 ∈ ((Cn‘ℝ)‘0) → ((ℝ D𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)) |
56 | 52, 55 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)) |
57 | 32, 56 | eqeltrrd 2886 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ)) |
58 | cncffvrn 23193 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (dom 𝐹–cn→ℂ)) → (𝐹 ∈ (dom 𝐹–cn→ℝ) ↔ 𝐹:dom 𝐹⟶ℝ)) | |
59 | 4, 57, 58 | sylancr 587 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (dom 𝐹–cn→ℝ) ↔ 𝐹:dom 𝐹⟶ℝ)) |
60 | 16, 59 | mpbird 258 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℝ)) |
61 | rescncf 23192 | . . 3 ⊢ ((𝐴[,]𝐵) ⊆ dom 𝐹 → (𝐹 ∈ (dom 𝐹–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))) | |
62 | 10, 60, 61 | sylc 65 | . 2 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
63 | 1, 2, 9, 47, 62 | c1lip1 24281 | 1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 ∃wrex 3108 ⊆ wss 3865 {cpr 4480 class class class wbr 4968 dom cdm 5450 ran crn 5451 ↾ cres 5452 Fun wfun 6226 Fn wfn 6227 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ↑pm cpm 8264 ℂcc 10388 ℝcr 10389 0cc0 10390 1c1 10391 + caddc 10393 · cmul 10395 ≤ cle 10529 − cmin 10723 ℕ0cn0 11751 ℤ≥cuz 12097 [,]cicc 12595 abscabs 14431 –cn→ccncf 23171 D cdv 24148 D𝑛 cdvn 24149 Cnccpn 24150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-fi 8728 df-sup 8759 df-inf 8760 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-ioo 12596 df-ico 12598 df-icc 12599 df-fz 12747 df-fzo 12888 df-seq 13224 df-exp 13284 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-hom 16422 df-cco 16423 df-rest 16529 df-topn 16530 df-0g 16548 df-gsum 16549 df-topgen 16550 df-pt 16551 df-prds 16554 df-xrs 16608 df-qtop 16613 df-imas 16614 df-xps 16616 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-mulg 17986 df-cntz 18192 df-cmn 18639 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-mopn 20227 df-fbas 20228 df-fg 20229 df-cnfld 20232 df-top 21190 df-topon 21207 df-topsp 21229 df-bases 21242 df-cld 21315 df-ntr 21316 df-cls 21317 df-nei 21394 df-lp 21432 df-perf 21433 df-cn 21523 df-cnp 21524 df-haus 21611 df-cmp 21683 df-tx 21858 df-hmeo 22051 df-fil 22142 df-fm 22234 df-flim 22235 df-flf 22236 df-xms 22617 df-ms 22618 df-tms 22619 df-cncf 23173 df-limc 24151 df-dv 24152 df-dvn 24153 df-cpn 24154 |
This theorem is referenced by: c1lip3 24283 |
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