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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 11212 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
| 5 | 1nn0 12542 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
| 6 | elcpn 25970 | . . . . 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 692 | . . . 4 ⊢ (𝐹 ∈ ((𝓑C𝑛‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ))) |
| 8 | 7 | simplbi 497 | . . 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 8887 | . . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → Fun 𝐹) | |
| 12 | 9, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
| 13 | 12 | funfnd 6597 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
| 14 | c1lip2.rn | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) | |
| 15 | df-f 6565 | . . . . . . 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 11236 | . . . . . . . . 9 ⊢ ℂ ∈ V | |
| 18 | reex 11246 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
| 19 | 17, 18 | elpm2 8914 | . . . . . . . 8 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
| 20 | 19 | simprbi 496 | . . . . . . 7 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ) |
| 21 | 9, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ) |
| 22 | dvfre 25989 | . . . . . 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 12388 | . . . . . . . . . . 11 ⊢ (0 + 1) = 1 | |
| 25 | 24 | fveq2i 6909 | . . . . . . . . . 10 ⊢ ((ℝ D𝑛 𝐹)‘(0 + 1)) = ((ℝ D𝑛 𝐹)‘1) |
| 26 | 0nn0 12541 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0 | |
| 27 | dvnp1 25961 | . . . . . . . . . . . 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 1454 | . . . . . . . . . . 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 2791 | . . . . . . . . 9 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘1) = (ℝ D ((ℝ D𝑛 𝐹)‘0))) |
| 31 | dvn0 25960 | . . . . . . . . . . 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 7447 | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D ((ℝ D𝑛 𝐹)‘0)) = (ℝ D 𝐹)) |
| 34 | 30, 33 | eqtrd 2777 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D𝑛 𝐹)‘1) = (ℝ D 𝐹)) |
| 35 | 7 | simprbi 496 | . . . . . . . . 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 2842 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ)) |
| 38 | cncff 24919 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ) | |
| 39 | fdm 6745 | . . . . . . 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 6722 | . . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom 𝐹⟶ℝ)) |
| 42 | 23, 41 | mpbid 232 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom 𝐹⟶ℝ) |
| 43 | cncfcdm 24924 | . . . . 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 257 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ)) |
| 46 | rescncf 24923 | . . 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 4762 | . . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ} |
| 49 | 1eluzge0 12934 | . . . . . . . . 9 ⊢ 1 ∈ (ℤ≥‘0) | |
| 50 | cpnord 25971 | . . . . . . . . 9 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ0 ∧ 1 ∈ (ℤ≥‘0)) → ((𝓑C𝑛‘ℝ)‘1) ⊆ ((𝓑C𝑛‘ℝ)‘0)) | |
| 51 | 48, 26, 49, 50 | mp3an 1463 | . . . . . . . 8 ⊢ ((𝓑C𝑛‘ℝ)‘1) ⊆ ((𝓑C𝑛‘ℝ)‘0) |
| 52 | 51, 3 | sselid 3981 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C𝑛‘ℝ)‘0)) |
| 53 | elcpn 25970 | . . . . . . . . 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 692 | . . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C𝑛‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ((ℝ D𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))) |
| 55 | 54 | simprbi 496 | . . . . . . 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 2842 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ)) |
| 58 | cncfcdm 24924 | . . . . 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 257 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℝ)) |
| 61 | rescncf 24923 | . . 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 26036 | 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 {cpr 4628 class class class wbr 5143 dom cdm 5685 ran crn 5686 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑pm cpm 8867 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 ℕ0cn0 12526 ℤ≥cuz 12878 [,]cicc 13390 abscabs 15273 –cn→ccncf 24902 D cdv 25898 D𝑛 cdvn 25899 𝓑C𝑛ccpn 25900 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-dvn 25903 df-cpn 25904 |
| This theorem is referenced by: c1lip3 26038 |
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