dvbdfbdioo - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioo 43520

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 11071	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		dvbdfbdioo.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	rexrd 11071	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
6	2, 4	readdcld 11050	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
7	6	rehalfcld 12266	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
8		dvbdfbdioo.altb	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
9		avglt1 12257	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
10	2, 4, 9	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
11	8, 10	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < ((𝐴 + 𝐵) / 2))
12		avglt2 12258	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
13	2, 4, 12	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
14	8, 13	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) < 𝐵)
15	3, 5, 7, 11, 14	eliood 43085	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
16	1, 15	ffvelcdmd 6994	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
17	16	recnd 11049	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
18	17	abscld 15193	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
19	18	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
20		simplr 767	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝑎 ∈ ℝ)
21	4	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐵 ∈ ℝ)
22	2	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐴 ∈ ℝ)
23	21, 22	resubcld 11449	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝐵 − 𝐴) ∈ ℝ)
24	20, 23	remulcld 11051	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝑎 · (𝐵 − 𝐴)) ∈ ℝ)
25	19, 24	readdcld 11050	. . 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 724	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐴 < 𝐵)
27	1	ad2antrr 724	. . . 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 724	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
30		2fveq3 6809	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
31	30	breq1d 5091	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎))
32	31	cbvralvw 3222	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎)
33	32	biimpi 215	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎)
34	33	adantl 483	. . . 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 43519	. . 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 6809	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑦)))
38	37	breq1d 5091	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑏))
39	38	cbvralvw 3222	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑏)
40		breq2 5085	. . . . . 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 3171	. . . . 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 3566	. . 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 585	. 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 3156	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ∀wral 3062 ∃wrex 3071 class class class wbr 5081 dom cdm 5600 ⟶wf 6454 ‘cfv 6458 (class class class)co 7307 ℝcr 10916 + caddc 10920 · cmul 10922 < clt 11055 ≤ cle 11056 − cmin 11251 / cdiv 11678 2c2 12074 (,)cioo 13125 abscabs 14990 D cdv 25072

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 ax-addf 10996 ax-mulf 10997

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-2o 8329 df-er 8529 df-map 8648 df-pm 8649 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-fi 9214 df-sup 9245 df-inf 9246 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-q 12735 df-rp 12777 df-xneg 12894 df-xadd 12895 df-xmul 12896 df-ioo 13129 df-ico 13131 df-icc 13132 df-fz 13286 df-fzo 13429 df-seq 13768 df-exp 13829 df-hash 14091 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-hom 17031 df-cco 17032 df-rest 17178 df-topn 17179 df-0g 17197 df-gsum 17198 df-topgen 17199 df-pt 17200 df-prds 17203 df-xrs 17258 df-qtop 17263 df-imas 17264 df-xps 17266 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-mulg 18746 df-cntz 18968 df-cmn 19433 df-psmet 20634 df-xmet 20635 df-met 20636 df-bl 20637 df-mopn 20638 df-fbas 20639 df-fg 20640 df-cnfld 20643 df-top 22088 df-topon 22105 df-topsp 22127 df-bases 22141 df-cld 22215 df-ntr 22216 df-cls 22217 df-nei 22294 df-lp 22332 df-perf 22333 df-cn 22423 df-cnp 22424 df-haus 22511 df-cmp 22583 df-tx 22758 df-hmeo 22951 df-fil 23042 df-fm 23134 df-flim 23135 df-flf 23136 df-xms 23518 df-ms 23519 df-tms 23520 df-cncf 24086 df-limc 25075 df-dv 25076

This theorem is referenced by: ioodvbdlimc1lem2 43522 ioodvbdlimc2lem 43524

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