c1lip2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > c1lip2		Structured version Visualization version GIF version

Theorem c1lip2 26022

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.)

**Hypotheses**
Ref	Expression
c1lip2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
c1lip2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
c1lip2.f	⊢ (𝜑 → 𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘1))
c1lip2.rn	⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
c1lip2.dm	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)

**Assertion**
Ref	Expression
c1lip2	⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))

Distinct variable groups: 𝜑,𝑥,𝑦,𝑘 𝑥,𝐴,𝑦,𝑘 𝑥,𝐵,𝑦,𝑘 𝑥,𝐹,𝑦,𝑘

**Proof of Theorem c1lip2**
Step	Hyp	Ref	Expression
1		c1lip2.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		c1lip2.b	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		c1lip2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘1))
4		ax-resscn 11215	. . . . 5 ⊢ ℝ ⊆ ℂ
5		1nn0 12540	. . . . 5 ⊢ 1 ∈ ℕ₀
6		elcpn 25955	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℕ₀) → (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ))))
7	4, 5, 6	mp2an 690	. . . 4 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ)))
8	7	simplbi 496	. . 3 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘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 8876	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → Fun 𝐹)
12	9, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
13	12	funfnd 6590	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
14		c1lip2.rn	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
15		df-f 6558	. . . . . . 7 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ))
16	13, 14, 15	sylanbrc 581	. . . . . 6 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
17		cnex 11239	. . . . . . . . 9 ⊢ ℂ ∈ V
18		reex 11249	. . . . . . . . 9 ⊢ ℝ ∈ V
19	17, 18	elpm2 8903	. . . . . . . 8 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
20	19	simprbi 495	. . . . . . 7 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → dom 𝐹 ⊆ ℝ)
21	9, 20	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
22		dvfre 25974	. . . . . 6 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
23	16, 21, 22	syl2anc 582	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
24		0p1e1 12386	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
25	24	fveq2i 6904	. . . . . . . . . 10 ⊢ ((ℝ D^𝑛 𝐹)‘(0 + 1)) = ((ℝ D^𝑛 𝐹)‘1)
26		0nn0 12539	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
27		dvnp1 25946	. . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 0 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(0 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
28	4, 26, 27	mp3an13 1449	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → ((ℝ D^𝑛 𝐹)‘(0 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
29	9, 28	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(0 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
30	25, 29	eqtr3id 2780	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
31		dvn0 25945	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ)) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
32	4, 9, 31	sylancr 585	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
33	32	oveq2d 7440	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D ((ℝ D^𝑛 𝐹)‘0)) = (ℝ D 𝐹))
34	30, 33	eqtrd 2766	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D 𝐹))
35	7	simprbi 495	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘1) → ((ℝ D^𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ))
36	3, 35	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ))
37	34, 36	eqeltrrd 2827	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ))
38		cncff 24904	. . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ)
39		fdm 6737	. . . . . . 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 6714	. . . . 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 24909	. . . . 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 585	. . . 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 24908	. . 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 4771	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
49		1eluzge0 12928	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘0)
50		cpnord 25956	. . . . . . . . 9 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 1 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0))
51	48, 26, 49, 50	mp3an 1458	. . . . . . . 8 ⊢ ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0)
52	51, 3	sselid 3977	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0))
53		elcpn 25955	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ 0 ∈ ℕ₀) → (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))))
54	4, 26, 53	mp2an 690	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)))
55	54	simprbi 495	. . . . . . 7 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0) → ((ℝ D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))
56	52, 55	syl 17	. . . . . 6 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))
57	32, 56	eqeltrrd 2827	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
58		cncfcdm 24909	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (dom 𝐹–cn→ℂ)) → (𝐹 ∈ (dom 𝐹–cn→ℝ) ↔ 𝐹:dom 𝐹⟶ℝ))
59	4, 57, 58	sylancr 585	. . . 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 24908	. . 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 26021	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 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ⊆ wss 3947 {cpr 4635 class class class wbr 5153 dom cdm 5682 ran crn 5683 ↾ cres 5684 Fun wfun 6548 Fn wfn 6549 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ↑_pm cpm 8856 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 ≤ cle 11299 − cmin 11494 ℕ₀cn0 12524 ℤ_≥cuz 12874 [,]cicc 13381 abscabs 15239 –cn→ccncf 24887 D cdv 25883 D^𝑛 cdvn 25884 𝓑C^𝑛ccpn 25885

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-cmp 23382 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 df-dvn 25888 df-cpn 25889

This theorem is referenced by: c1lip3 26023

W3C validator