cnplimc - Metamath Proof Explorer

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Theorem cnplimc 26055

Description: A function is continuous at 𝐵 iff its limit at 𝐵 equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
cnplimc.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
cnplimc.j	⊢ 𝐽 = (𝐾 ↾_t 𝐴)

**Assertion**
Ref	Expression
cnplimc	⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵))))

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

Step	Hyp	Ref	Expression
1		cnplimc.j	. . . . 5 ⊢ 𝐽 = (𝐾 ↾_t 𝐴)
2		cnplimc.k	. . . . . . 7 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24948	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		simpl 487	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ℂ)
5		resttopon 23327	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
6	3, 4, 5	sylancr 598	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
7	1, 6	eqeltrid 2867	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝐴))
8		cnpf2 23416	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐹:𝐴⟶ℂ)
9	8	3expia 1139	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ))
10	7, 3, 9	sylancl 597	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ))
11	10	pm4.71rd 571	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))))
12		simpr 489	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:𝐴⟶ℂ)
13		simplr 780	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ 𝐴)
14	13	snssd 4752	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → {𝐵} ⊆ 𝐴)
15		ssequn2 4142	. . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐵}) = 𝐴)
16	14, 15	sylib 221	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐴 ∪ {𝐵}) = 𝐴)
17	16	feq2d 6689	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹:(𝐴 ∪ {𝐵})⟶ℂ ↔ 𝐹:𝐴⟶ℂ))
18	12, 17	mpbird 260	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:(𝐴 ∪ {𝐵})⟶ℂ)
19	18	feqmptd 6949	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹 = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)))
20	16	oveq2d 7426	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t 𝐴))
21	1, 20	eqtr4id 2817	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐽 = (𝐾 ↾_t (𝐴 ∪ {𝐵})))
22	21	oveq1d 7425	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐽 CnP 𝐾) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾))
23	22	fveq1d 6883	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → ((𝐽 CnP 𝐾)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
24	19, 23	eleq12d 2857	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
25		eqid 2763	. . . . 5 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
26		ifid 4528	. . . . . . 7 ⊢ if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = (𝐹‘𝑥)
27		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
28	27	adantl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∪ {𝐵}) ∧ 𝑥 = 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐵))
29	28	ifeq1da 4519	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
30	26, 29	eqtr3id 2812	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → (𝐹‘𝑥) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
31	30	mpteq2ia 5206	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
32		simpll 778	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐴 ⊆ ℂ)
33	32, 13	sseldd 3938	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ ℂ)
34	25, 2, 31, 12, 32, 33	ellimc 26041	. . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → ((𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
35	24, 34	bitr4d 285	. . 3 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵)))
36	35	pm5.32da 589	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵))))
37	11, 36	bitrd 282	1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∪ cun 3903 ⊆ wss 3905 ifcif 4487 {csn 4589 ↦ cmpt 5192 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11102 ↾_t crest 17477 TopOpenctopn 17478 ℂ_fldccnfld 21531 TopOnctopon 23076 CnP ccnp 23391 lim_ℂ climc 26030

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9367 df-sup 9398 df-inf 9399 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-fz 13540 df-seq 14043 df-exp 14103 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-struct 17211 df-slot 17246 df-ndx 17258 df-base 17274 df-plusg 17327 df-mulr 17328 df-starv 17329 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-rest 17479 df-topn 17480 df-topgen 17500 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cnp 23394 df-xms 24486 df-ms 24487 df-limc 26034

This theorem is used by: cnlimc 26056 dvcnp2 26088 dvmulbr 26107 dvcobr 26114 cncfiooicclem1 46635 jumpncnp 46640 dirkercncf 46849 fourierdlem32 46881 fourierdlem33 46882 fourierdlem62 46910 fouriercnp 46968

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