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 25162

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 10928	. . . . 5 ⊢ ℝ ⊆ ℂ
5		1nn0 12249	. . . . 5 ⊢ 1 ∈ ℕ₀
6		elcpn 25098	. . . . 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 689	. . . 4 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ)))
8	7	simplbi 498	. . 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 8635	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → Fun 𝐹)
12	9, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
13	12	funfnd 6465	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
14		c1lip2.rn	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
15		df-f 6437	. . . . . . 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 10952	. . . . . . . . 9 ⊢ ℂ ∈ V
18		reex 10962	. . . . . . . . 9 ⊢ ℝ ∈ V
19	17, 18	elpm2 8662	. . . . . . . 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 25115	. . . . . 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 12095	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
25	24	fveq2i 6777	. . . . . . . . . 10 ⊢ ((ℝ D^𝑛 𝐹)‘(0 + 1)) = ((ℝ D^𝑛 𝐹)‘1)
26		0nn0 12248	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
27		dvnp1 25089	. . . . . . . . . . . 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 1451	. . . . . . . . . . 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 2792	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
31		dvn0 25088	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ)) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
32	4, 9, 31	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
33	32	oveq2d 7291	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D ((ℝ D^𝑛 𝐹)‘0)) = (ℝ D 𝐹))
34	30, 33	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D 𝐹))
35	7	simprbi 497	. . . . . . . . 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 2840	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ))
38		cncff 24056	. . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ)
39		fdm 6609	. . . . . . 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 6586	. . . . 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		cncffvrn 24061	. . . . 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 24060	. . 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 4698	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
49		1eluzge0 12632	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘0)
50		cpnord 25099	. . . . . . . . 9 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 1 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0))
51	48, 26, 49, 50	mp3an 1460	. . . . . . . 8 ⊢ ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0)
52	51, 3	sselid 3919	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0))
53		elcpn 25098	. . . . . . . . 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 689	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)))
55	54	simprbi 497	. . . . . . 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 2840	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
58		cncffvrn 24061	. . . . 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 24060	. . 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 25161	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 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 {cpr 4563 class class class wbr 5074 dom cdm 5589 ran crn 5590 ↾ cres 5591 Fun wfun 6427 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_pm cpm 8616 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 ℕ₀cn0 12233 ℤ_≥cuz 12582 [,]cicc 13082 abscabs 14945 –cn→ccncf 24039 D cdv 25027 D^𝑛 cdvn 25028 𝓑C^𝑛ccpn 25029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-dvn 25032 df-cpn 25033

This theorem is referenced by: c1lip3 25163

W3C validator