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Theorem c1lip3 26039

Description: C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
c1lip3.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
c1lip3.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
c1lip3.f	⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ ((𝓑C^𝑛‘ℝ)‘1))
c1lip3.rn	⊢ (𝜑 → (𝐹 “ ℝ) ⊆ ℝ)
c1lip3.dm	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)

**Assertion**
Ref	Expression
c1lip3	⊢ (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 c1lip3**
Step	Hyp	Ref	Expression
1		c1lip3.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		c1lip3.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		c1lip3.f	. . 3 ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ ((𝓑C^𝑛‘ℝ)‘1))
4		df-ima 5697	. . . 4 ⊢ (𝐹 “ ℝ) = ran (𝐹 ↾ ℝ)
5		c1lip3.rn	. . . 4 ⊢ (𝜑 → (𝐹 “ ℝ) ⊆ ℝ)
6	4, 5	eqsstrrid 4022	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ ℝ) ⊆ ℝ)
7		iccssre 13470	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
8	1, 2, 7	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
9		c1lip3.dm	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)
10	8, 9	ssind 4240	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (ℝ ∩ dom 𝐹))
11		dmres 6029	. . . 4 ⊢ dom (𝐹 ↾ ℝ) = (ℝ ∩ dom 𝐹)
12	10, 11	sseqtrrdi 4024	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom (𝐹 ↾ ℝ))
13	1, 2, 3, 6, 12	c1lip2 26038	. 2 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
14	8	sseld 3981	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
15	8	sseld 3981	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ))
16	14, 15	anim12d 609	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
17	16	imp 406	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
18		fvres 6924	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → ((𝐹 ↾ ℝ)‘𝑦) = (𝐹‘𝑦))
19		fvres 6924	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → ((𝐹 ↾ ℝ)‘𝑥) = (𝐹‘𝑥))
20	18, 19	oveqan12rd 7452	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥)) = ((𝐹‘𝑦) − (𝐹‘𝑥)))
21	20	fveq2d 6909	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (abs‘(((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
22	21	breq1d 5152	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((abs‘(((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
23	22	biimpd 229	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((abs‘(((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
24	17, 23	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((abs‘(((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
25	24	ralimdvva 3205	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
26	25	reximdv 3169	. 2 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((𝐹 ↾ ℝ)‘𝑦) − ((𝐹 ↾ ℝ)‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
27	13, 26	mpd 15	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 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ∩ cin 3949 ⊆ wss 3950 class class class wbr 5142 dom cdm 5684 ran crn 5685 ↾ cres 5686 “ cima 5687 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 1c1 11157 · cmul 11161 ≤ cle 11297 − cmin 11493 [,]cicc 13391 abscabs 15274 𝓑C^𝑛ccpn 25901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-dvn 25904 df-cpn 25905

This theorem is referenced by: aalioulem3 26377

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