ioodvbdlimc2 - Mathbox for Glauco Siliprandi

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Theorem ioodvbdlimc2 40468

Description: A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)

**Hypotheses**
Ref	Expression
ioodvbdlimc2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ioodvbdlimc2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ioodvbdlimc2.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
ioodvbdlimc2.dmdv	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
ioodvbdlimc2.dvbd	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)

**Assertion**
Ref	Expression
ioodvbdlimc2	⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) ≠ ∅)

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑥,𝐹,𝑦 𝜑,𝑥,𝑦

Proof of Theorem ioodvbdlimc2 Dummy variables 𝑗 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioodvbdlimc2.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ)
3		ioodvbdlimc2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ)
5		simpr 476	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
6		ioodvbdlimc2.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
8		ioodvbdlimc2.dmdv	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
10		ioodvbdlimc2.dvbd	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)
12		fveq2 6229	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥))
13	12	fveq2d 6233	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (abs‘((ℝ D 𝐹)‘𝑦)) = (abs‘((ℝ D 𝐹)‘𝑥)))
14	13	cbvmptv 4783	. . . . . 6 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
15	14	rneqi 5384	. . . . 5 ⊢ ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))) = ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
16	15	supeq1i 8394	. . . 4 ⊢ sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )
17		eqid 2651	. . . 4 ⊢ ((⌊‘(1 / (𝐵 − 𝐴))) + 1) = ((⌊‘(1 / (𝐵 − 𝐴))) + 1)
18		oveq2 6698	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (1 / 𝑘) = (1 / 𝑗))
19	18	oveq2d 6706	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 − (1 / 𝑘)) = (𝐵 − (1 / 𝑗)))
20	19	fveq2d 6233	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝐵 − (1 / 𝑘))) = (𝐹‘(𝐵 − (1 / 𝑗))))
21	20	cbvmptv 4783	. . . 4 ⊢ (𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘)))) = (𝑗 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑗))))
22	19	cbvmptv 4783	. . . 4 ⊢ (𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐵 − (1 / 𝑘))) = (𝑗 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐵 − (1 / 𝑗)))
23		eqid 2651	. . . 4 ⊢ if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1)) = if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1))
24		biid 251	. . . 4 ⊢ (((((((𝜑 ∧ 𝐴 < 𝐵) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ (ℤ_≥‘if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1)))) ∧ (abs‘(((𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))‘𝑗) − (lim sup‘(𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐵)) < (1 / 𝑗)) ↔ ((((((𝜑 ∧ 𝐴 < 𝐵) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ (ℤ_≥‘if(((⌊‘(1 / (𝐵 − 𝐴))) + 1) ≤ ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(sup(ran (𝑦 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑦))), ℝ, < ) / (𝑥 / 2))) + 1), ((⌊‘(1 / (𝐵 − 𝐴))) + 1)))) ∧ (abs‘(((𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))‘𝑗) − (lim sup‘(𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐵)) < (1 / 𝑗)))
25	2, 4, 5, 7, 9, 11, 16, 17, 21, 22, 23, 24	ioodvbdlimc2lem 40467	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (lim sup‘(𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))) ∈ (𝐹 lim_ℂ 𝐵))
26		ne0i 3954	. . 3 ⊢ ((lim sup‘(𝑘 ∈ (ℤ_≥‘((⌊‘(1 / (𝐵 − 𝐴))) + 1)) ↦ (𝐹‘(𝐵 − (1 / 𝑘))))) ∈ (𝐹 lim_ℂ 𝐵) → (𝐹 lim_ℂ 𝐵) ≠ ∅)
27	25, 26	syl 17	. 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (𝐹 lim_ℂ 𝐵) ≠ ∅)
28		ax-resscn 10031	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ)
30	6, 29	fssd 6095	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
32		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴)
33	1	rexrd 10127	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ^*)
35	3	rexrd 10127	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ^*)
37		ioo0 12238	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
38	34, 36, 37	syl2anc 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
39	32, 38	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐴(,)𝐵) = ∅)
40	39	feq2d 6069	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐹:(𝐴(,)𝐵)⟶ℂ ↔ 𝐹:∅⟶ℂ))
41	31, 40	mpbid 222	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐹:∅⟶ℂ)
42	3	recnd 10106	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	42	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℂ)
44	41, 43	limcdm0 40168	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐹 lim_ℂ 𝐵) = ℂ)
45		0cn 10070	. . . . 5 ⊢ 0 ∈ ℂ
46	45	ne0ii 3956	. . . 4 ⊢ ℂ ≠ ∅
47	46	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ℂ ≠ ∅)
48	44, 47	eqnetrd 2890	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → (𝐹 lim_ℂ 𝐵) ≠ ∅)
49	27, 48, 1, 3	ltlecasei 10183	1 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 ∃wrex 2942 ⊆ wss 3607 ∅c0 3948 ifcif 4119 class class class wbr 4685 ↦ cmpt 4762 dom cdm 5143 ran crn 5144 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 supcsup 8387 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 ℝ^*cxr 10111 < clt 10112 ≤ cle 10113 − cmin 10304 / cdiv 10722 2c2 11108 ℤ_≥cuz 11725 ℝ⁺crp 11870 (,)cioo 12213 ⌊cfl 12631 abscabs 14018 lim supclsp 14245 lim_ℂ climc 23671 D cdv 23672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-cmp 21238 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676

This theorem is referenced by: fourierdlem94 40735 fourierdlem113 40754

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