icocncflimc - Mathbox for Glauco Siliprandi

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Theorem icocncflimc 45090

Description: Limit at the lower bound, of a continuous function defined on a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
icocncflimc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
icocncflimc.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
icocncflimc.altb	⊢ (𝜑 → 𝐴 < 𝐵)
icocncflimc.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ))

**Assertion**
Ref	Expression
icocncflimc	⊢ (𝜑 → (𝐹‘𝐴) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))

Proof of Theorem icocncflimc Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icocncflimc.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ))
2		icocncflimc.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	rexrd 11261	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		icocncflimc.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5	2	leidd 11777	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
6		icocncflimc.altb	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
7	3, 4, 3, 5, 6	elicod 13371	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,)𝐵))
8	1, 7	cnlimci 25740	. 2 ⊢ (𝜑 → (𝐹‘𝐴) ∈ (𝐹 lim_ℂ 𝐴))
9		cncfrss 24733	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ) → (𝐴[,)𝐵) ⊆ ℂ)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ℂ)
11		ssid 3996	. . . . . . 7 ⊢ ℂ ⊆ ℂ
12		eqid 2724	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13		eqid 2724	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵))
14		eqid 2724	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
15	12, 13, 14	cncfcn 24752	. . . . . . 7 ⊢ (((𝐴[,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
16	10, 11, 15	sylancl 585	. . . . . 6 ⊢ (𝜑 → ((𝐴[,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
17	1, 16	eleqtrd 2827	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
18	12	cnfldtopon 24621	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
20		resttopon 22987	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴[,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ∈ (TopOn‘(𝐴[,)𝐵)))
21	19, 10, 20	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ∈ (TopOn‘(𝐴[,)𝐵)))
22	12	cnfldtop 24622	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
23		unicntop 24624	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
24	23	restid 17378	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
25	22, 24	ax-mp 5	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
26	25	cnfldtopon 24621	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) ∈ (TopOn‘ℂ)
27		cncnp 23106	. . . . . 6 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ∈ (TopOn‘(𝐴[,)𝐵)) ∧ ((TopOpen‘ℂ_fld) ↾_t ℂ) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)) ↔ (𝐹:(𝐴[,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴[,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) CnP ((TopOpen‘ℂ_fld) ↾_t ℂ))‘𝑥))))
28	21, 26, 27	sylancl 585	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)) ↔ (𝐹:(𝐴[,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴[,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) CnP ((TopOpen‘ℂ_fld) ↾_t ℂ))‘𝑥))))
29	17, 28	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐹:(𝐴[,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴[,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) CnP ((TopOpen‘ℂ_fld) ↾_t ℂ))‘𝑥)))
30	29	simpld 494	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,)𝐵)⟶ℂ)
31		ioossico 13412	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)
32	31	a1i 11	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵))
33		eqid 2724	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴}))
34	2	recnd 11239	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
35	23	ntrtop 22896	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((int‘(TopOpen‘ℂ_fld))‘ℂ) = ℂ)
36	22, 35	ax-mp 5	. . . . . . . 8 ⊢ ((int‘(TopOpen‘ℂ_fld))‘ℂ) = ℂ
37		undif 4473	. . . . . . . . . . 11 ⊢ ((𝐴[,)𝐵) ⊆ ℂ ↔ ((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))) = ℂ)
38	10, 37	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))) = ℂ)
39	38	eqcomd 2730	. . . . . . . . 9 ⊢ (𝜑 → ℂ = ((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))))
40	39	fveq2d 6885	. . . . . . . 8 ⊢ (𝜑 → ((int‘(TopOpen‘ℂ_fld))‘ℂ) = ((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))))
41	36, 40	eqtr3id 2778	. . . . . . 7 ⊢ (𝜑 → ℂ = ((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))))
42	34, 41	eleqtrd 2827	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))))
43	42, 7	elind 4186	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))) ∩ (𝐴[,)𝐵)))
44	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
45		ssid 3996	. . . . . . 7 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,)𝐵)
46	45	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ (𝐴[,)𝐵))
47	23, 13	restntr 23008	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴[,)𝐵) ⊆ ℂ ∧ (𝐴[,)𝐵) ⊆ (𝐴[,)𝐵)) → ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))‘(𝐴[,)𝐵)) = (((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))) ∩ (𝐴[,)𝐵)))
48	44, 10, 46, 47	syl3anc 1368	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))‘(𝐴[,)𝐵)) = (((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))) ∩ (𝐴[,)𝐵)))
49	43, 48	eleqtrrd 2828	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))‘(𝐴[,)𝐵)))
50	7	snssd 4804	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ (𝐴[,)𝐵))
51		ssequn2 4175	. . . . . . . . 9 ⊢ ({𝐴} ⊆ (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
52	50, 51	sylib 217	. . . . . . . 8 ⊢ (𝜑 → ((𝐴[,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
53	52	eqcomd 2730	. . . . . . 7 ⊢ (𝜑 → (𝐴[,)𝐵) = ((𝐴[,)𝐵) ∪ {𝐴}))
54	53	oveq2d 7417	. . . . . 6 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴})))
55	54	fveq2d 6885	. . . . 5 ⊢ (𝜑 → (int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵))) = (int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴}))))
56		snunioo1 44710	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
57	3, 4, 6, 56	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
58	57	eqcomd 2730	. . . . 5 ⊢ (𝜑 → (𝐴[,)𝐵) = ((𝐴(,)𝐵) ∪ {𝐴}))
59	55, 58	fveq12d 6888	. . . 4 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))‘(𝐴[,)𝐵)) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴})))‘((𝐴(,)𝐵) ∪ {𝐴})))
60	49, 59	eleqtrd 2827	. . 3 ⊢ (𝜑 → 𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴})))‘((𝐴(,)𝐵) ∪ {𝐴})))
61	30, 32, 10, 12, 33, 60	limcres 25737	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴) = (𝐹 lim_ℂ 𝐴))
62	8, 61	eleqtrrd 2828	1 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 {csn 4620 class class class wbr 5138 ↾ cres 5668 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 ℝcr 11105 ℝ^*cxr 11244 < clt 11245 (,)cioo 13321 [,)cico 13323 ↾_t crest 17365 TopOpenctopn 17366 ℂ_fldccnfld 21228 Topctop 22717 TopOnctopon 22734 intcnt 22843 Cn ccn 23050 CnP ccnp 23051 –cn→ccncf 24718 lim_ℂ climc 25713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17367 df-topn 17368 df-topgen 17388 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-ntr 22846 df-cn 23053 df-cnp 23054 df-xms 24148 df-ms 24149 df-cncf 24720 df-limc 25717

This theorem is referenced by: fourierdlem46 45353

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