icocncflimc - Mathbox for Glauco Siliprandi

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Theorem icocncflimc 44592

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 25398	. 2 ⊢ (𝜑 → (𝐹‘𝐴) ∈ (𝐹 lim_ℂ 𝐴))
9		cncfrss 24399	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ) → (𝐴[,)𝐵) ⊆ ℂ)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ℂ)
11		ssid 4004	. . . . . . 7 ⊢ ℂ ⊆ ℂ
12		eqid 2733	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13		eqid 2733	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵))
14		eqid 2733	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
15	12, 13, 14	cncfcn 24418	. . . . . . 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 587	. . . . . 6 ⊢ (𝜑 → ((𝐴[,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
17	1, 16	eleqtrd 2836	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
18	12	cnfldtopon 24291	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
20		resttopon 22657	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴[,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ∈ (TopOn‘(𝐴[,)𝐵)))
21	19, 10, 20	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ∈ (TopOn‘(𝐴[,)𝐵)))
22	12	cnfldtop 24292	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
23		unicntop 24294	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
24	23	restid 17376	. . . . . . . 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 24291	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) ∈ (TopOn‘ℂ)
27		cncnp 22776	. . . . . 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 587	. . . . 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 496	. . 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 2733	. . 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 22566	. . . . . . . . 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 4481	. . . . . . . . . . 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 2739	. . . . . . . . 9 ⊢ (𝜑 → ℂ = ((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))))
40	39	fveq2d 6893	. . . . . . . 8 ⊢ (𝜑 → ((int‘(TopOpen‘ℂ_fld))‘ℂ) = ((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))))
41	36, 40	eqtr3id 2787	. . . . . . 7 ⊢ (𝜑 → ℂ = ((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))))
42	34, 41	eleqtrd 2836	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((int‘(TopOpen‘ℂ_fld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))))
43	42, 7	elind 4194	. . . . 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 4004	. . . . . . 7 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,)𝐵)
46	45	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ (𝐴[,)𝐵))
47	23, 13	restntr 22678	. . . . . 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 1372	. . . . 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 2837	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))‘(𝐴[,)𝐵)))
50	7	snssd 4812	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ (𝐴[,)𝐵))
51		ssequn2 4183	. . . . . . . . 9 ⊢ ({𝐴} ⊆ (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
52	50, 51	sylib 217	. . . . . . . 8 ⊢ (𝜑 → ((𝐴[,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
53	52	eqcomd 2739	. . . . . . 7 ⊢ (𝜑 → (𝐴[,)𝐵) = ((𝐴[,)𝐵) ∪ {𝐴}))
54	53	oveq2d 7422	. . . . . 6 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴})))
55	54	fveq2d 6893	. . . . 5 ⊢ (𝜑 → (int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵))) = (int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴}))))
56		snunioo1 44212	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
57	3, 4, 6, 56	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
58	57	eqcomd 2739	. . . . 5 ⊢ (𝜑 → (𝐴[,)𝐵) = ((𝐴(,)𝐵) ∪ {𝐴}))
59	55, 58	fveq12d 6896	. . . 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 2836	. . 3 ⊢ (𝜑 → 𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,)𝐵) ∪ {𝐴})))‘((𝐴(,)𝐵) ∪ {𝐴})))
61	30, 32, 10, 12, 33, 60	limcres 25395	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴) = (𝐹 lim_ℂ 𝐴))
62	8, 61	eleqtrrd 2837	1 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 {csn 4628 class class class wbr 5148 ↾ cres 5678 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 ℝ^*cxr 11244 < clt 11245 (,)cioo 13321 [,)cico 13323 ↾_t crest 17363 TopOpenctopn 17364 ℂ_fldccnfld 20937 Topctop 22387 TopOnctopon 22404 intcnt 22513 Cn ccn 22720 CnP ccnp 22721 –cn→ccncf 24384 lim_ℂ climc 25371

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 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 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-mulr 17208 df-starv 17209 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-rest 17365 df-topn 17366 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-ntr 22516 df-cn 22723 df-cnp 22724 df-xms 23818 df-ms 23819 df-cncf 24386 df-limc 25375

This theorem is referenced by: fourierdlem46 44855

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