cnplimc - Metamath Proof Explorer

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Theorem cnplimc 23651

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 22586	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		simpl 473	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ℂ)
5		resttopon 20965	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
6	3, 4, 5	sylancr 695	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
7	1, 6	syl5eqel 2705	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝐴))
8		cnpf2 21054	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐹:𝐴⟶ℂ)
9	8	3expia 1267	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ))
10	7, 3, 9	sylancl 694	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ))
11	10	pm4.71rd 667	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))))
12		simpr 477	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:𝐴⟶ℂ)
13		simplr 792	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ 𝐴)
14	13	snssd 4340	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → {𝐵} ⊆ 𝐴)
15		ssequn2 3786	. . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐵}) = 𝐴)
16	14, 15	sylib 208	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐴 ∪ {𝐵}) = 𝐴)
17	16	feq2d 6031	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹:(𝐴 ∪ {𝐵})⟶ℂ ↔ 𝐹:𝐴⟶ℂ))
18	12, 17	mpbird 247	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:(𝐴 ∪ {𝐵})⟶ℂ)
19	18	feqmptd 6249	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹 = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)))
20	16	oveq2d 6666	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t 𝐴))
21	20, 1	syl6reqr 2675	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐽 = (𝐾 ↾_t (𝐴 ∪ {𝐵})))
22	21	oveq1d 6665	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐽 CnP 𝐾) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾))
23	22	fveq1d 6193	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → ((𝐽 CnP 𝐾)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
24	19, 23	eleq12d 2695	. . . 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 2622	. . . . 5 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
26		ifid 4125	. . . . . . 7 ⊢ if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = (𝐹‘𝑥)
27		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
28	27	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∪ {𝐵}) ∧ 𝑥 = 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐵))
29	28	ifeq1da 4116	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
30	26, 29	syl5eqr 2670	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → (𝐹‘𝑥) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
31	30	mpteq2ia 4740	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
32		simpll 790	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐴 ⊆ ℂ)
33	32, 13	sseldd 3604	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ ℂ)
34	25, 2, 31, 12, 32, 33	ellimc 23637	. . . 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 271	. . 3 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵)))
36	35	pm5.32da 673	. 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 268	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

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 ifcif 4086 {csn 4177 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 TopOnctopon 20715 CnP ccnp 21029 lim_ℂ climc 23626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cnp 21032 df-xms 22125 df-ms 22126 df-limc 23630

This theorem is referenced by: cnlimc 23652 dvcnp2 23683 dvmulbr 23702 dvcobr 23709 cncfiooicclem1 40106 jumpncnp 40111 dirkercncf 40324 fourierdlem32 40356 fourierdlem33 40357 fourierdlem62 40385 fouriercnp 40443

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