c1lip2 - Metamath Proof Explorer

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Theorem c1lip2 25931

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 11063	. . . . 5 ⊢ ℝ ⊆ ℂ
5		1nn0 12397	. . . . 5 ⊢ 1 ∈ ℕ₀
6		elcpn 25864	. . . . 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 8771	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → Fun 𝐹)
12	9, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
13	12	funfnd 6512	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
14		c1lip2.rn	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
15		df-f 6485	. . . . . . 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 11087	. . . . . . . . 9 ⊢ ℂ ∈ V
18		reex 11097	. . . . . . . . 9 ⊢ ℝ ∈ V
19	17, 18	elpm2 8798	. . . . . . . 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 25883	. . . . . 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 12242	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
25	24	fveq2i 6825	. . . . . . . . . 10 ⊢ ((ℝ D^𝑛 𝐹)‘(0 + 1)) = ((ℝ D^𝑛 𝐹)‘1)
26		0nn0 12396	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
27		dvnp1 25855	. . . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘1) = (ℝ D ((ℝ D^𝑛 𝐹)‘0)))
31		dvn0 25854	. . . . . . . . . . 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 7362	. . . . . . . . 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 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 2832	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ))
38		cncff 24814	. . . . . . 7 ⊢ ((ℝ D 𝐹) ∈ (dom 𝐹–cn→ℂ) → (ℝ D 𝐹):dom 𝐹⟶ℂ)
39		fdm 6660	. . . . . . 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 6635	. . . . 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 24819	. . . . 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 24818	. . 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 4715	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
49		1eluzge0 12778	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘0)
50		cpnord 25865	. . . . . . . . 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 3932	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝓑C^𝑛‘ℝ)‘0))
53		elcpn 25864	. . . . . . . . 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 2832	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
58		cncfcdm 24819	. . . . 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 24818	. . 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 25930	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 2111 ∀wral 3047 ∃wrex 3056 ⊆ wss 3902 {cpr 4578 class class class wbr 5091 dom cdm 5616 ran crn 5617 ↾ cres 5618 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ↑_pm cpm 8751 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 ≤ cle 11147 − cmin 11344 ℕ₀cn0 12381 ℤ_≥cuz 12732 [,]cicc 13248 abscabs 15141 –cn→ccncf 24797 D cdv 25792 D^𝑛 cdvn 25793 𝓑C^𝑛ccpn 25794

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19230 df-cmn 19695 df-psmet 21284 df-xmet 21285 df-met 21286 df-bl 21287 df-mopn 21288 df-fbas 21289 df-fg 21290 df-cnfld 21293 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-cmp 23303 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796 df-dvn 25797 df-cpn 25798

This theorem is referenced by: c1lip3 25932

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