Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dvcnp | Structured version Visualization version GIF version |
Description: The difference quotient is continuous at 𝐵 when the original function is differentiable at 𝐵. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
dvcnp.j | ⊢ 𝐽 = (𝐾 ↾t 𝐴) |
dvcnp.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
dvcnp.g | ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) |
Ref | Expression |
---|---|
dvcnp | ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcnp.g | . 2 ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) | |
2 | dvfg 24431 | . . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
3 | 2 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
4 | ffun 6510 | . . . . . 6 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
5 | funfvbrb 6813 | . . . . . 6 ⊢ (Fun (𝑆 D 𝐹) → (𝐵 ∈ dom (𝑆 D 𝐹) ↔ 𝐵(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐵))) | |
6 | 3, 4, 5 | 3syl 18 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝐵 ∈ dom (𝑆 D 𝐹) ↔ 𝐵(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐵))) |
7 | eqid 2818 | . . . . . 6 ⊢ (𝐾 ↾t 𝑆) = (𝐾 ↾t 𝑆) | |
8 | dvcnp.k | . . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
9 | eqid 2818 | . . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) = (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) | |
10 | recnprss 24429 | . . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
11 | 10 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝑆 ⊆ ℂ) |
12 | simp2 1129 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐹:𝐴⟶ℂ) | |
13 | simp3 1130 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐴 ⊆ 𝑆) | |
14 | 7, 8, 9, 11, 12, 13 | eldv 24423 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝐵(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐵) ↔ (𝐵 ∈ ((int‘(𝐾 ↾t 𝑆))‘𝐴) ∧ ((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) limℂ 𝐵)))) |
15 | 6, 14 | bitrd 280 | . . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝐵 ∈ dom (𝑆 D 𝐹) ↔ (𝐵 ∈ ((int‘(𝐾 ↾t 𝑆))‘𝐴) ∧ ((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) limℂ 𝐵)))) |
16 | 15 | simplbda 500 | . . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) limℂ 𝐵)) |
17 | 13, 11 | sstrd 3974 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐴 ⊆ ℂ) |
18 | 17 | adantr 481 | . . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐴 ⊆ ℂ) |
19 | 11, 12, 13 | dvbss 24426 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → dom (𝑆 D 𝐹) ⊆ 𝐴) |
20 | 19 | sselda 3964 | . . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐵 ∈ 𝐴) |
21 | eldifsn 4711 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵)) | |
22 | 12 | adantr 481 | . . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹:𝐴⟶ℂ) |
23 | 22, 18, 20 | dvlem 24421 | . . . . 5 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)) ∈ ℂ) |
24 | 21, 23 | sylan2br 594 | . . . 4 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵)) → (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)) ∈ ℂ) |
25 | dvcnp.j | . . . 4 ⊢ 𝐽 = (𝐾 ↾t 𝐴) | |
26 | 18, 20, 24, 25, 8 | limcmpt2 24409 | . . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → (((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) limℂ 𝐵) ↔ (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) ∈ ((𝐽 CnP 𝐾)‘𝐵))) |
27 | 16, 26 | mpbid 233 | . 2 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) ∈ ((𝐽 CnP 𝐾)‘𝐵)) |
28 | 1, 27 | eqeltrid 2914 | 1 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 ⊆ wss 3933 ifcif 4463 {csn 4557 {cpr 4559 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 − cmin 10858 / cdiv 11285 ↾t crest 16682 TopOpenctopn 16683 ℂfldccnfld 20473 intcnt 21553 CnP ccnp 21761 limℂ climc 24387 D cdv 24388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-topn 16685 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cnp 21764 df-haus 21851 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-limc 24391 df-dv 24392 |
This theorem is referenced by: efrlim 25474 |
Copyright terms: Public domain | W3C validator |