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Mirrors > Home > MPE Home > Th. List > eldv | Structured version Visualization version GIF version |
Description: The differentiable predicate. A function 𝐹 is differentiable at 𝐵 with derivative 𝐶 iff 𝐹 is defined in a neighborhood of 𝐵 and the difference quotient has limit 𝐶 at 𝐵. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
dvval.t | ⊢ 𝑇 = (𝐾 ↾t 𝑆) |
dvval.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
eldv.g | ⊢ 𝐺 = (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) |
eldv.s | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
eldv.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
eldv.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
Ref | Expression |
---|---|
eldv | ⊢ (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 limℂ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldv.s | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
2 | eldv.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | eldv.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | dvval.t | . . . . . 6 ⊢ 𝑇 = (𝐾 ↾t 𝑆) | |
5 | dvval.k | . . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
6 | 4, 5 | dvfval 25396 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ))) |
7 | 1, 2, 3, 6 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ))) |
8 | 7 | simpld 496 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))) |
9 | 8 | eleq2d 2820 | . 2 ⊢ (𝜑 → (〈𝐵, 𝐶〉 ∈ (𝑆 D 𝐹) ↔ 〈𝐵, 𝐶〉 ∈ ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)))) |
10 | df-br 5148 | . . 3 ⊢ (𝐵(𝑆 D 𝐹)𝐶 ↔ 〈𝐵, 𝐶〉 ∈ (𝑆 D 𝐹)) | |
11 | 10 | bicomi 223 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ (𝑆 D 𝐹) ↔ 𝐵(𝑆 D 𝐹)𝐶) |
12 | sneq 4637 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) | |
13 | 12 | difeq2d 4121 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝐵})) |
14 | fveq2 6888 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) | |
15 | 14 | oveq2d 7420 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑧) − (𝐹‘𝑥)) = ((𝐹‘𝑧) − (𝐹‘𝐵))) |
16 | oveq2 7412 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑧 − 𝑥) = (𝑧 − 𝐵)) | |
17 | 15, 16 | oveq12d 7422 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) |
18 | 13, 17 | mpteq12dv 5238 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) |
19 | eldv.g | . . . . 5 ⊢ 𝐺 = (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) | |
20 | 18, 19 | eqtr4di 2791 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = 𝐺) |
21 | id 22 | . . . 4 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵) | |
22 | 20, 21 | oveq12d 7422 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥) = (𝐺 limℂ 𝐵)) |
23 | 22 | opeliunxp2 5836 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 limℂ 𝐵))) |
24 | 9, 11, 23 | 3bitr3g 313 | 1 ⊢ (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 limℂ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 〈cop 4633 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 − cmin 11440 / cdiv 11867 ↾t crest 17362 TopOpenctopn 17363 ℂfldccnfld 20929 intcnt 22503 limℂ climc 25361 D cdv 25362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cnp 22714 df-xms 23808 df-ms 23809 df-limc 25365 df-dv 25366 |
This theorem is referenced by: dvcl 25398 perfdvf 25402 dvreslem 25408 dvres2lem 25409 dvidlem 25414 dvcnp 25418 dvcnp2 25419 dvaddbr 25437 dvmulbr 25438 dvcobr 25445 dvcjbr 25448 dvrec 25454 dvcnvlem 25475 dveflem 25478 dvferm1 25484 dvferm2 25486 ftc1 25541 taylthlem1 25867 ulmdvlem3 25896 gg-dvcnp2 35112 gg-dvmulbr 35113 gg-dvcobr 35114 unbdqndv1 35322 ftc1cnnc 36498 fperdvper 44570 |
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