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Theorem c1lip2 24589

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 10588	. . . . 5 ⊢ ℝ ⊆ ℂ
5		1nn0 11907	. . . . 5 ⊢ 1 ∈ ℕ₀
6		elcpn 24525	. . . . 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 500	. . 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 8420	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → Fun 𝐹)
12	9, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
13	12	funfnd 6381	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
14		c1lip2.rn	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
15		df-f 6354	. . . . . . 7 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ))
16	13, 14, 15	sylanbrc 585	. . . . . 6 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
17		cnex 10612	. . . . . . . . 9 ⊢ ℂ ∈ V
18		reex 10622	. . . . . . . . 9 ⊢ ℝ ∈ V
19	17, 18	elpm2 8432	. . . . . . . 8 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
20	19	simprbi 499	. . . . . . 7 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → dom 𝐹 ⊆ ℝ)
21	9, 20	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
22		dvfre 24542	. . . . . 6 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
23	16, 21, 22	syl2anc 586	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
24		0p1e1 11753	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
25	24	fveq2i 6668	. . . . . . . . . 10 ⊢ ((ℝ D^𝑛 𝐹)‘(0 + 1)) = ((ℝ D^𝑛 𝐹)‘1)
26		0nn0 11906	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
27		dvnp1 24516	. . . . . . . . . . . 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 1448	. . . . . . . . . . 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	syl5eqr 2870	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
31		dvn0 24515	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ)) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
32	4, 9, 31	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
33	32	oveq2d 7166	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D ((ℝ D^𝑛 𝐹)‘0)) = (ℝ D 𝐹))
34	30, 33	eqtrd 2856	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D 𝐹))
35	7	simprbi 499	. . . . . . . . 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 2914	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ))
38		cncff 23495	. . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ)
39		fdm 6517	. . . . . . 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 6495	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom 𝐹⟶ℝ))
42	23, 41	mpbid 234	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom 𝐹⟶ℝ)
43		cncffvrn 23500	. . . . 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 589	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ) ↔ (ℝ D 𝐹):dom 𝐹⟶ℝ))
45	42, 44	mpbird 259	. . 3 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℝ))
46		rescncf 23499	. . 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 4692	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
49		1eluzge0 12286	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘0)
50		cpnord 24526	. . . . . . . . 9 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 1 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0))
51	48, 26, 49, 50	mp3an 1457	. . . . . . . 8 ⊢ ((𝓑C^𝑛‘ℝ)‘1) ⊆ ((𝓑C^𝑛‘ℝ)‘0)
52	51, 3	sseldi 3965	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0))
53		elcpn 24525	. . . . . . . . 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 499	. . . . . . 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 2914	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
58		cncffvrn 23500	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (dom 𝐹–cn→ℂ)) → (𝐹 ∈ (dom 𝐹–cn→ℝ) ↔ 𝐹:dom 𝐹⟶ℝ))
59	4, 57, 58	sylancr 589	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (dom 𝐹–cn→ℝ) ↔ 𝐹:dom 𝐹⟶ℝ))
60	16, 59	mpbird 259	. . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℝ))
61		rescncf 23499	. . 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 24588	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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 {cpr 4563 class class class wbr 5059 dom cdm 5550 ran crn 5551 ↾ cres 5552 Fun wfun 6344 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ↑_pm cpm 8401 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 ≤ cle 10670 − cmin 10864 ℕ₀cn0 11891 ℤ_≥cuz 12237 [,]cicc 12735 abscabs 14587 –cn→ccncf 23478 D cdv 24455 D^𝑛 cdvn 24456 𝓑C^𝑛ccpn 24457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-dvn 24460 df-cpn 24461

This theorem is referenced by: c1lip3 24590

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