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| Mirrors > Home > MPE Home > Th. List > cnplimc | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| cnplimc.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
| cnplimc.j | ⊢ 𝐽 = (𝐾 ↾t 𝐴) |
| Ref | Expression |
|---|---|
| cnplimc | ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnplimc.j | . . . . 5 ⊢ 𝐽 = (𝐾 ↾t 𝐴) | |
| 2 | cnplimc.k | . . . . . . 7 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
| 3 | 2 | cnfldtopon 24803 | . . . . . 6 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
| 4 | simpl 482 | . . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ℂ) | |
| 5 | resttopon 23169 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → (𝐾 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
| 6 | 3, 4, 5 | sylancr 587 | . . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐾 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
| 7 | 1, 6 | eqeltrid 2845 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝐴)) |
| 8 | cnpf2 23258 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐹:𝐴⟶ℂ) | |
| 9 | 8 | 3expia 1122 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ)) |
| 10 | 7, 3, 9 | sylancl 586 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐹:𝐴⟶ℂ)) |
| 11 | 10 | pm4.71rd 562 | . 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)))) |
| 12 | simpr 484 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:𝐴⟶ℂ) | |
| 13 | simplr 769 | . . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ 𝐴) | |
| 14 | 13 | snssd 4809 | . . . . . . . . 9 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → {𝐵} ⊆ 𝐴) |
| 15 | ssequn2 4189 | . . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐵}) = 𝐴) | |
| 16 | 14, 15 | sylib 218 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐴 ∪ {𝐵}) = 𝐴) |
| 17 | 16 | feq2d 6722 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹:(𝐴 ∪ {𝐵})⟶ℂ ↔ 𝐹:𝐴⟶ℂ)) |
| 18 | 12, 17 | mpbird 257 | . . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹:(𝐴 ∪ {𝐵})⟶ℂ) |
| 19 | 18 | feqmptd 6977 | . . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐹 = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥))) |
| 20 | 16 | oveq2d 7447 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐾 ↾t (𝐴 ∪ {𝐵})) = (𝐾 ↾t 𝐴)) |
| 21 | 1, 20 | eqtr4id 2796 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐽 = (𝐾 ↾t (𝐴 ∪ {𝐵}))) |
| 22 | 21 | oveq1d 7446 | . . . . . 6 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐽 CnP 𝐾) = ((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)) |
| 23 | 22 | fveq1d 6908 | . . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → ((𝐽 CnP 𝐾)‘𝐵) = (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)) |
| 24 | 19, 23 | eleq12d 2835 | . . . 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 2737 | . . . . 5 ⊢ (𝐾 ↾t (𝐴 ∪ {𝐵})) = (𝐾 ↾t (𝐴 ∪ {𝐵})) | |
| 26 | ifid 4566 | . . . . . . 7 ⊢ if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = (𝐹‘𝑥) | |
| 27 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) | |
| 28 | 27 | adantl 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∪ {𝐵}) ∧ 𝑥 = 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐵)) |
| 29 | 28 | ifeq1da 4557 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → if(𝑥 = 𝐵, (𝐹‘𝑥), (𝐹‘𝑥)) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥))) |
| 30 | 26, 29 | eqtr3id 2791 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → (𝐹‘𝑥) = if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥))) |
| 31 | 30 | mpteq2ia 5245 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐹‘𝐵), (𝐹‘𝑥))) |
| 32 | simpll 767 | . . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐴 ⊆ ℂ) | |
| 33 | 32, 13 | sseldd 3984 | . . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → 𝐵 ∈ ℂ) |
| 34 | 25, 2, 31, 12, 32, 33 | ellimc 25908 | . . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → ((𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐹‘𝑥)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))) |
| 35 | 24, 34 | bitr4d 282 | . . 3 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))) |
| 36 | 35 | pm5.32da 579 | . 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
| 37 | 11, 36 | bitrd 279 | 1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ifcif 4525 {csn 4626 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ↾t crest 17465 TopOpenctopn 17466 ℂfldccnfld 21364 TopOnctopon 22916 CnP ccnp 23233 limℂ climc 25897 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cnp 23236 df-xms 24330 df-ms 24331 df-limc 25901 |
| This theorem is referenced by: cnlimc 25923 dvcnp2 25955 dvcnp2OLD 25956 dvmulbr 25975 dvmulbrOLD 25976 dvcobr 25983 dvcobrOLD 25984 cncfiooicclem1 45908 jumpncnp 45913 dirkercncf 46122 fourierdlem32 46154 fourierdlem33 46155 fourierdlem62 46183 fouriercnp 46241 |
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