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Mirrors > Home > MPE Home > Th. List > cnlimc | Structured version Visualization version GIF version |
Description: 𝐹 is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
cnlimc | ⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3923 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | eqid 2737 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
3 | eqid 2737 | . . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴) | |
4 | 2 | cnfldtopon 23680 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
5 | 4 | toponrestid 21818 | . . . . 5 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
6 | 2, 3, 5 | cncfcn 23807 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld))) |
7 | 1, 6 | mpan2 691 | . . 3 ⊢ (𝐴 ⊆ ℂ → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld))) |
8 | 7 | eleq2d 2823 | . 2 ⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) ↔ 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))) |
9 | resttopon 22058 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
10 | 4, 9 | mpan 690 | . . 3 ⊢ (𝐴 ⊆ ℂ → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴)) |
11 | cncnp 22177 | . . 3 ⊢ ((((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥)))) | |
12 | 10, 4, 11 | sylancl 589 | . 2 ⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥)))) |
13 | 2, 3 | cnplimc 24784 | . . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥)))) |
14 | 13 | baibd 543 | . . . . 5 ⊢ (((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥))) |
15 | 14 | an32s 652 | . . . 4 ⊢ (((𝐴 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥))) |
16 | 15 | ralbidva 3117 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) → (∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥))) |
17 | 16 | pm5.32da 582 | . 2 ⊢ (𝐴 ⊆ ℂ → ((𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥)) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥)))) |
18 | 8, 12, 17 | 3bitrd 308 | 1 ⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ⊆ wss 3866 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ↾t crest 16925 TopOpenctopn 16926 ℂfldccnfld 20363 TopOnctopon 21807 Cn ccn 22121 CnP ccnp 22122 –cn→ccncf 23773 limℂ climc 24759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-fz 13096 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-mulr 16816 df-starv 16817 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-rest 16927 df-topn 16928 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cn 22124 df-cnp 22125 df-xms 23218 df-ms 23219 df-cncf 23775 df-limc 24763 |
This theorem is referenced by: cnlimci 24786 fourierdlem62 43384 |
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