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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 ∈ ((𝓑C𝑛‘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 ∈ ((𝓑C𝑛‘ℝ)‘1)) | |
4 | ax-resscn 11021 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
5 | 1nn0 12342 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
6 | elcpn 25196 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ)))) | |
7 | 4, 5, 6 | mp2an 689 | . . . 4 ⊢ (𝐹 ∈ ((𝓑C𝑛‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ))) |
8 | 7 | simplbi 498 | . . 3 ⊢ (𝐹 ∈ ((𝓑C𝑛‘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 8698 | . . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → Fun 𝐹) | |
12 | 9, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
13 | 12 | funfnd 6509 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
14 | c1lip2.rn | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) | |
15 | df-f 6477 | . . . . . . 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 11045 | . . . . . . . . 9 ⊢ ℂ ∈ V | |
18 | reex 11055 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
19 | 17, 18 | elpm2 8725 | . . . . . . . 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 25213 | . . . . . 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 12188 | . . . . . . . . . . 11 ⊢ (0 + 1) = 1 | |
25 | 24 | fveq2i 6822 | . . . . . . . . . 10 ⊢ ((ℝ D𝑛 𝐹)‘(0 + 1)) = ((ℝ D𝑛 𝐹)‘1) |
26 | 0nn0 12341 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0 | |
27 | dvnp1 25187 | . . . . . . . . . . . 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 1451 | . . . . . . . . . . 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 | eqtr3id 2790 | . . . . . . . . 9 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘1) = (ℝ D ((ℝ D𝑛 𝐹)‘0))) |
31 | dvn0 25186 | . . . . . . . . . . 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 7345 | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D ((ℝ D𝑛 𝐹)‘0)) = (ℝ D 𝐹)) |
34 | 30, 33 | eqtrd 2776 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘1) = (ℝ D 𝐹)) |
35 | 7 | simprbi 497 | . . . . . . . . 9 ⊢ (𝐹 ∈ ((𝓑C𝑛‘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 2838 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ)) |
38 | cncff 24154 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ) | |
39 | fdm 6654 | . . . . . . 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 6631 | . . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom 𝐹⟶ℝ)) |
42 | 23, 41 | mpbid 231 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom 𝐹⟶ℝ) |
43 | cncfcdm 24159 | . . . . 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 256 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ)) |
46 | rescncf 24158 | . . 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 4709 | . . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ} |
49 | 1eluzge0 12725 | . . . . . . . . 9 ⊢ 1 ∈ (ℤ≥‘0) | |
50 | cpnord 25197 | . . . . . . . . 9 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ0 ∧ 1 ∈ (ℤ≥‘0)) → ((𝓑C𝑛‘ℝ)‘1) ⊆ ((𝓑C𝑛‘ℝ)‘0)) | |
51 | 48, 26, 49, 50 | mp3an 1460 | . . . . . . . 8 ⊢ ((𝓑C𝑛‘ℝ)‘1) ⊆ ((𝓑C𝑛‘ℝ)‘0) |
52 | 51, 3 | sselid 3929 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C𝑛‘ℝ)‘0)) |
53 | elcpn 25196 | . . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ 0 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)))) | |
54 | 4, 26, 53 | mp2an 689 | . . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C𝑛‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))) |
55 | 54 | simprbi 497 | . . . . . . 7 ⊢ (𝐹 ∈ ((𝓑C𝑛‘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 2838 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ)) |
58 | cncfcdm 24159 | . . . . 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 256 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℝ)) |
61 | rescncf 24158 | . . 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 25259 | 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 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 ∃wrex 3070 ⊆ wss 3897 {cpr 4574 class class class wbr 5089 dom cdm 5614 ran crn 5615 ↾ cres 5616 Fun wfun 6467 Fn wfn 6468 ⟶wf 6469 ‘cfv 6473 (class class class)co 7329 ↑pm cpm 8679 ℂcc 10962 ℝcr 10963 0cc0 10964 1c1 10965 + caddc 10967 · cmul 10969 ≤ cle 11103 − cmin 11298 ℕ0cn0 12326 ℤ≥cuz 12675 [,]cicc 13175 abscabs 15036 –cn→ccncf 24137 D cdv 25125 D𝑛 cdvn 25126 𝓑C𝑛ccpn 25127 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 ax-addf 11043 ax-mulf 11044 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-2o 8360 df-er 8561 df-map 8680 df-pm 8681 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-fi 9260 df-sup 9291 df-inf 9292 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-q 12782 df-rp 12824 df-xneg 12941 df-xadd 12942 df-xmul 12943 df-ioo 13176 df-ico 13178 df-icc 13179 df-fz 13333 df-fzo 13476 df-seq 13815 df-exp 13876 df-hash 14138 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-starv 17066 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-unif 17074 df-hom 17075 df-cco 17076 df-rest 17222 df-topn 17223 df-0g 17241 df-gsum 17242 df-topgen 17243 df-pt 17244 df-prds 17247 df-xrs 17302 df-qtop 17307 df-imas 17308 df-xps 17310 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-mulg 18789 df-cntz 19011 df-cmn 19475 df-psmet 20687 df-xmet 20688 df-met 20689 df-bl 20690 df-mopn 20691 df-fbas 20692 df-fg 20693 df-cnfld 20696 df-top 22141 df-topon 22158 df-topsp 22180 df-bases 22194 df-cld 22268 df-ntr 22269 df-cls 22270 df-nei 22347 df-lp 22385 df-perf 22386 df-cn 22476 df-cnp 22477 df-haus 22564 df-cmp 22636 df-tx 22811 df-hmeo 23004 df-fil 23095 df-fm 23187 df-flim 23188 df-flf 23189 df-xms 23571 df-ms 23572 df-tms 23573 df-cncf 24139 df-limc 25128 df-dv 25129 df-dvn 25130 df-cpn 25131 |
This theorem is referenced by: c1lip3 25261 |
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