dvbdfbdioo - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioo 46358

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 11195	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		dvbdfbdioo.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	rexrd 11195	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
6	2, 4	readdcld 11174	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
7	6	rehalfcld 12424	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
8		dvbdfbdioo.altb	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
9		avglt1 12415	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < ((𝐴 + 𝐵) / 2))
12		avglt2 12416	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) < 𝐵)
15	3, 5, 7, 11, 14	eliood 45928	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
16	1, 15	ffvelcdmd 7037	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
17	16	recnd 11173	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
18	17	abscld 15401	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
19	18	ad2antrr 727	. . . 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 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐵 ∈ ℝ)
22	2	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐴 ∈ ℝ)
23	21, 22	resubcld 11578	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝐵 − 𝐴) ∈ ℝ)
24	20, 23	remulcld 11175	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → (𝑎 · (𝐵 − 𝐴)) ∈ ℝ)
25	19, 24	readdcld 11174	. . 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 727	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → 𝐴 < 𝐵)
27	1	ad2antrr 727	. . . 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 727	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
30		2fveq3 6845	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
31	30	breq1d 5095	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝑎))
32	31	cbvralvw 3215	. . . . . 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 46357	. . 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 6845	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑦)))
38	37	breq1d 5095	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑏))
39	38	cbvralvw 3215	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑏)
40		breq2 5089	. . . . . 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 3160	. . . . 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 3564	. . 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 3145	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 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 class class class wbr 5085 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 2c2 12236 (,)cioo 13298 abscabs 15196 D cdv 25830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834

This theorem is referenced by: ioodvbdlimc1lem2 46360 ioodvbdlimc2lem 46362

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