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Mirrors > Home > MPE Home > Th. List > cnlimci | Structured version Visualization version GIF version |
Description: If 𝐹 is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
cnlimci.f | ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐷)) |
cnlimci.c | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
cnlimci | ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6670 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) | |
2 | oveq2 7164 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐹 limℂ 𝑥) = (𝐹 limℂ 𝐵)) | |
3 | 1, 2 | eleq12d 2907 | . 2 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥) ↔ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))) |
4 | cnlimci.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐷)) | |
5 | cncfrss 23499 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐷) → 𝐴 ⊆ ℂ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
7 | cncfrss2 23500 | . . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐷) → 𝐷 ⊆ ℂ) | |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
9 | ssid 3989 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
10 | cncfss 23507 | . . . . 5 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐴–cn→𝐷) ⊆ (𝐴–cn→ℂ)) | |
11 | 8, 9, 10 | sylancl 588 | . . . 4 ⊢ (𝜑 → (𝐴–cn→𝐷) ⊆ (𝐴–cn→ℂ)) |
12 | 11, 4 | sseldd 3968 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) |
13 | cnlimc 24486 | . . . 4 ⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥)))) | |
14 | 13 | simplbda 502 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→ℂ)) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥)) |
15 | 6, 12, 14 | syl2anc 586 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥)) |
16 | cnlimci.c | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
17 | 3, 15, 16 | rspcdva 3625 | 1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 –cn→ccncf 23484 limℂ climc 24460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-xms 22930 df-ms 22931 df-cncf 23486 df-limc 24464 |
This theorem is referenced by: cnmptlimc 24488 dvcnvlem 24573 ioccncflimc 42188 icocncflimc 42192 dirkercncflem2 42409 fourierdlem84 42495 fourierdlem85 42496 fourierdlem88 42499 fourierdlem111 42522 fouriercn 42537 |
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