dvbdfbdioo - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioo 45914

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 11318	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		dvbdfbdioo.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	rexrd 11318	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
6	2, 4	readdcld 11297	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
7	6	rehalfcld 12520	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
8		dvbdfbdioo.altb	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
9		avglt1 12511	. . . . . . . . . . 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 12512	. . . . . . . . . . 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 45480	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
16	1, 15	ffvelcdmd 7112	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
17	16	recnd 11296	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
18	17	abscld 15481	. . . . 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 769	. . . . 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 11698	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝐵 − 𝐴) ∈ ℝ)
24	20, 23	remulcld 11298	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝑎 · (𝐵 − 𝐴)) ∈ ℝ)
25	19, 24	readdcld 11297	. . 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 6919	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
31	30	breq1d 5161	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎))
32	31	cbvralvw 3237	. . . . . 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 2737	. . . 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 45913	. . 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 6919	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑦)))
38	37	breq1d 5161	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑏))
39	38	cbvralvw 3237	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑏)
40		breq2 5155	. . . . . 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 3178	. . . . 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 3625	. . 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 3162	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 1539 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 class class class wbr 5151 dom cdm 5693 ⟶wf 6565 ‘cfv 6569 (class class class)co 7438 ℝcr 11161 + caddc 11165 · cmul 11167 < clt 11302 ≤ cle 11303 − cmin 11499 / cdiv 11927 2c2 12328 (,)cioo 13393 abscabs 15279 D cdv 25924

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239 ax-pre-sup 11240 ax-addf 11241

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-se 5646 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-isom 6578 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-of 7704 df-om 7895 df-1st 8022 df-2nd 8023 df-supp 8194 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-2o 8515 df-er 8753 df-map 8876 df-pm 8877 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-fsupp 9409 df-fi 9458 df-sup 9489 df-inf 9490 df-oi 9557 df-card 9986 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-div 11928 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-q 12998 df-rp 13042 df-xneg 13161 df-xadd 13162 df-xmul 13163 df-ioo 13397 df-ico 13399 df-icc 13400 df-fz 13554 df-fzo 13701 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-struct 17190 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-mulr 17321 df-starv 17322 df-sca 17323 df-vsca 17324 df-ip 17325 df-tset 17326 df-ple 17327 df-ds 17329 df-unif 17330 df-hom 17331 df-cco 17332 df-rest 17478 df-topn 17479 df-0g 17497 df-gsum 17498 df-topgen 17499 df-pt 17500 df-prds 17503 df-xrs 17558 df-qtop 17563 df-imas 17564 df-xps 17566 df-mre 17640 df-mrc 17641 df-acs 17643 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21383 df-xmet 21384 df-met 21385 df-bl 21386 df-mopn 21387 df-fbas 21388 df-fg 21389 df-cnfld 21392 df-top 22925 df-topon 22942 df-topsp 22964 df-bases 22978 df-cld 23052 df-ntr 23053 df-cls 23054 df-nei 23131 df-lp 23169 df-perf 23170 df-cn 23260 df-cnp 23261 df-haus 23348 df-cmp 23420 df-tx 23595 df-hmeo 23788 df-fil 23879 df-fm 23971 df-flim 23972 df-flf 23973 df-xms 24355 df-ms 24356 df-tms 24357 df-cncf 24929 df-limc 25927 df-dv 25928

This theorem is referenced by: ioodvbdlimc1lem2 45916 ioodvbdlimc2lem 45918

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