cnplimc - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cnplimc		Structured version Visualization version GIF version

Theorem cnplimc 26222

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 25116	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		simpl 488	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ℂ)
5		resttopon 23494	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
6	3, 4, 5	sylancr 599	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
7	1, 6	eqeltrid 2868	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝐴))
8		cnpf2 23583	. . . . 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 598	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ))
11	10	pm4.71rd 572	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))))
12		simpr 490	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:𝐴⟶ℂ)
13		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ 𝐴)
14	13	snssd 4747	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → {𝐵} ⊆ 𝐴)
15		ssequn2 4190	. . . . . . . . 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 6704	. . . . . . 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 6964	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹 = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)))
20	16	oveq2d 7446	. . . . . . . 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 7445	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐽 CnP 𝐾) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾))
23	22	fveq1d 6898	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → ((𝐽 CnP 𝐾)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
24	19, 23	eleq12d 2858	. . . 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 4526	. . . . . . 7 ⊢ if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = (𝐹‘𝑥)
27		fveq2 6896	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
28	27	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∪ {𝐵}) ∧ 𝑥 = 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐵))
29	28	ifeq1da 4517	. . . . . . 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 5203	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
32		simpll 779	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐴 ⊆ ℂ)
33	32, 13	sseldd 3914	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ ℂ)
34	25, 2, 31, 12, 32, 33	ellimc 26208	. . . 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 590	. 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 401 = wceq 1569 ∈ wcel 2146 ⊆ wss 3900 ∪ cun 4152 ifcif 4485 {csn 4588 ↦ cmpt 5189 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 ℂcc 11211 ↾_t crest 17606 TopOpenctopn 17607 ℂ_fldccnfld 21693 TopOnctopon 23243 CnP ccnp 23558 lim_ℂ climc 26197

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-er 8731 df-map 8863 df-pm 8864 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-fi 9416 df-sup 9447 df-inf 9448 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-z 12707 df-dec 12828 df-uz 12979 df-q 13089 df-rp 13134 df-xneg 13254 df-xadd 13255 df-xmul 13256 df-fz 13653 df-seq 14158 df-exp 14218 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-struct 17340 df-slot 17375 df-ndx 17387 df-base 17403 df-plusg 17456 df-mulr 17457 df-starv 17458 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-rest 17608 df-topn 17609 df-topgen 17629 df-psmet 21685 df-xmet 21686 df-met 21687 df-bl 21688 df-mopn 21689 df-cnfld 21694 df-top 23227 df-topon 23244 df-topsp 23266 df-bases 23279 df-cnp 23561 df-xms 24654 df-ms 24655 df-limc 26201

This theorem is used by: cnlimc 26223 dvcnp2 26255 dvmulbr 26274 dvcobr 26281 cncfiooicclem1 47045 jumpncnp 47050 dirkercncf 47259 fourierdlem32 47291 fourierdlem33 47292 fourierdlem62 47320 fouriercnp 47378

W3C validator