dvbdfbdioo - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioo 46170

Description: A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dvbdfbdioo.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvbdfbdioo.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvbdfbdioo.altb	⊢ (𝜑 → 𝐴 < 𝐵)
dvbdfbdioo.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
dvbdfbdioo.dmdv	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvbdfbdioo.dvbd	⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎)

**Assertion**
Ref	Expression
dvbdfbdioo	⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏)

Distinct variable groups: 𝐴,𝑎,𝑏,𝑥 𝐵,𝑎,𝑏,𝑥 𝐹,𝑎,𝑏,𝑥 𝜑,𝑎 Allowed substitution hints: 𝜑(𝑥,𝑏)

Proof of Theorem dvbdfbdioo Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvbdfbdioo.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
2		dvbdfbdioo.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	rexrd 11182	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		dvbdfbdioo.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	rexrd 11182	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
6	2, 4	readdcld 11161	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
7	6	rehalfcld 12388	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
8		dvbdfbdioo.altb	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
9		avglt1 12379	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
10	2, 4, 9	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
11	8, 10	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < ((𝐴 + 𝐵) / 2))
12		avglt2 12380	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
13	2, 4, 12	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
14	8, 13	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) < 𝐵)
15	3, 5, 7, 11, 14	eliood 45740	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
16	1, 15	ffvelcdmd 7030	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
17	16	recnd 11160	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
18	17	abscld 15362	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
19	18	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
20		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝑎 ∈ ℝ)
21	4	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐵 ∈ ℝ)
22	2	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐴 ∈ ℝ)
23	21, 22	resubcld 11565	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝐵 − 𝐴) ∈ ℝ)
24	20, 23	remulcld 11162	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝑎 · (𝐵 − 𝐴)) ∈ ℝ)
25	19, 24	readdcld 11161	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴))) ∈ ℝ)
26	8	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐴 < 𝐵)
27	1	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
28		dvbdfbdioo.dmdv	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
29	28	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
30		2fveq3 6839	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
31	30	breq1d 5108	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎))
32	31	cbvralvw 3214	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎)
33	32	biimpi 216	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎)
34	33	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎)
35		eqid 2736	. . . 4 ⊢ ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴))) = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴)))
36	22, 21, 26, 27, 29, 20, 34, 35	dvbdfbdioolem2 46169	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴))))
37		2fveq3 6839	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑦)))
38	37	breq1d 5108	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑏))
39	38	cbvralvw 3214	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑏)
40		breq2 5102	. . . . . 6 ⊢ (𝑏 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴))) → ((abs‘(𝐹‘𝑦)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑦)) ≤ ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴)))))
41	40	ralbidv 3159	. . . . 5 ⊢ (𝑏 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴))) → (∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑏 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴)))))
42	39, 41	bitrid 283	. . . 4 ⊢ (𝑏 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴))) → (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴)))))
43	42	rspcev 3576	. . 3 ⊢ ((((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴))) ∈ ℝ ∧ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝑎 · (𝐵 − 𝐴)))) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏)
44	25, 36, 43	syl2anc 584	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏)
45		dvbdfbdioo.dvbd	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎)
46	44, 45	r19.29a 3144	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 class class class wbr 5098 dom cdm 5624 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 + caddc 11029 · cmul 11031 < clt 11166 ≤ cle 11167 − cmin 11364 / cdiv 11794 2c2 12200 (,)cioo 13261 abscabs 15157 D cdv 25820

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: ioodvbdlimc1lem2 46172 ioodvbdlimc2lem 46174

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