cnplimc - Metamath Proof Explorer

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Theorem cnplimc 25873

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 24766	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		simpl 483	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ℂ)
5		resttopon 23145	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
6	3, 4, 5	sylancr 593	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐾 ↾_t 𝐴) ∈ (TopOn‘𝐴))
7	1, 6	eqeltrid 2843	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝐴))
8		cnpf2 23234	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐹:𝐴⟶ℂ)
9	8	3expia 1127	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ))
10	7, 3, 9	sylancl 592	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ))
11	10	pm4.71rd 567	. 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))))
12		simpr 485	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:𝐴⟶ℂ)
13		simplr 774	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ 𝐴)
14	13	snssd 4719	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → {𝐵} ⊆ 𝐴)
15		ssequn2 4119	. . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐵}) = 𝐴)
16	14, 15	sylib 219	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐴 ∪ {𝐵}) = 𝐴)
17	16	feq2d 6640	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹:(𝐴 ∪ {𝐵})⟶ℂ ↔ 𝐹:𝐴⟶ℂ))
18	12, 17	mpbird 258	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:(𝐴 ∪ {𝐵})⟶ℂ)
19	18	feqmptd 6896	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹 = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)))
20	16	oveq2d 7373	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t 𝐴))
21	1, 20	eqtr4id 2793	. . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐽 = (𝐾 ↾_t (𝐴 ∪ {𝐵})))
22	21	oveq1d 7372	. . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐽 CnP 𝐾) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾))
23	22	fveq1d 6830	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → ((𝐽 CnP 𝐾)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
24	19, 23	eleq12d 2833	. . . 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 2739	. . . . 5 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
26		ifid 4496	. . . . . . 7 ⊢ if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = (𝐹‘𝑥)
27		fveq2 6828	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
28	27	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∪ {𝐵}) ∧ 𝑥 = 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐵))
29	28	ifeq1da 4487	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
30	26, 29	eqtr3id 2788	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → (𝐹‘𝑥) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
31	30	mpteq2ia 5168	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥)))
32		simpll 772	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐴 ⊆ ℂ)
33	32, 13	sseldd 3916	. . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ ℂ)
34	25, 2, 31, 12, 32, 33	ellimc 25859	. . . 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 283	. . 3 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵)))
36	35	pm5.32da 584	. 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 280	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 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∪ cun 3881 ⊆ wss 3883 ifcif 4455 {csn 4556 ↦ cmpt 5154 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ℂcc 11028 ↾_t crest 17375 TopOpenctopn 17376 ℂ_fldccnfld 21348 TopOnctopon 22894 CnP ccnp 23209 lim_ℂ climc 25848

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-map 8766 df-pm 8767 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9315 df-sup 9346 df-inf 9347 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-fz 13454 df-seq 13956 df-exp 14016 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17172 df-plusg 17225 df-mulr 17226 df-starv 17227 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-rest 17377 df-topn 17378 df-topgen 17398 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cnp 23212 df-xms 24304 df-ms 24305 df-limc 25852

This theorem is referenced by: cnlimc 25874 dvcnp2 25906 dvmulbr 25925 dvcobr 25932 cncfiooicclem1 46344 jumpncnp 46349 dirkercncf 46558 fourierdlem32 46590 fourierdlem33 46591 fourierdlem62 46619 fouriercnp 46677

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