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 25950

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 11074	. . . . 5 ⊢ ℝ ⊆ ℂ
5		1nn0 12408	. . . . 5 ⊢ 1 ∈ ℕ₀
6		elcpn 25883	. . . . 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 692	. . . 4 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘1) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘1) ∈ (dom 𝐹–cn→ℂ)))
8	7	simplbi 497	. . 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 8780	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → Fun 𝐹)
12	9, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
13	12	funfnd 6520	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
14		c1lip2.rn	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
15		df-f 6493	. . . . . . 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 11098	. . . . . . . . 9 ⊢ ℂ ∈ V
18		reex 11108	. . . . . . . . 9 ⊢ ℝ ∈ V
19	17, 18	elpm2 8808	. . . . . . . 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 25902	. . . . . 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 12253	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
25	24	fveq2i 6834	. . . . . . . . . 10 ⊢ ((ℝ D^𝑛 𝐹)‘(0 + 1)) = ((ℝ D^𝑛 𝐹)‘1)
26		0nn0 12407	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
27		dvnp1 25874	. . . . . . . . . . . 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 1454	. . . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
31		dvn0 25873	. . . . . . . . . . 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 7371	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D ((ℝ D^𝑛 𝐹)‘0)) = (ℝ D 𝐹))
34	30, 33	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D 𝐹))
35	7	simprbi 496	. . . . . . . . 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 2834	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ))
38		cncff 24833	. . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ)
39		fdm 6668	. . . . . . 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 6643	. . . . 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 24838	. . . . 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 24837	. . 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 4716	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
49		1eluzge0 12784	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘0)
50		cpnord 25884	. . . . . . . . 9 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 1 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0))
51	48, 26, 49, 50	mp3an 1463	. . . . . . . 8 ⊢ ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0)
52	51, 3	sselid 3928	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0))
53		elcpn 25883	. . . . . . . . 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 692	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ((ℝ D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)))
55	54	simprbi 496	. . . . . . 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 2834	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
58		cncfcdm 24838	. . . . 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 24837	. . 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 25949	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 1541 ∈ wcel 2113 ∀wral 3048 ∃wrex 3057 ⊆ wss 3898 {cpr 4579 class class class wbr 5095 dom cdm 5621 ran crn 5622 ↾ cres 5623 Fun wfun 6483 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ↑_pm cpm 8760 ℂcc 11015 ℝcr 11016 0cc0 11017 1c1 11018 + caddc 11020 · cmul 11022 ≤ cle 11158 − cmin 11355 ℕ₀cn0 12392 ℤ_≥cuz 12742 [,]cicc 13255 abscabs 15148 –cn→ccncf 24816 D cdv 25811 D^𝑛 cdvn 25812 𝓑C^𝑛ccpn 25813

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-mulg 18989 df-cntz 19237 df-cmn 19702 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-fbas 21297 df-fg 21298 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-lp 23071 df-perf 23072 df-cn 23162 df-cnp 23163 df-haus 23250 df-cmp 23322 df-tx 23497 df-hmeo 23690 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-xms 24255 df-ms 24256 df-tms 24257 df-cncf 24818 df-limc 25814 df-dv 25815 df-dvn 25816 df-cpn 25817

This theorem is referenced by: c1lip3 25951

W3C validator