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Mirrors > Home > MPE Home > Th. List > dvbssntr | Structured version Visualization version GIF version |
Description: The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
dvcl.s | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
dvcl.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
dvcl.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
dvbssntr.j | ⊢ 𝐽 = (𝐾 ↾t 𝑆) |
dvbssntr.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
dvbssntr | ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
2 | dvcl.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | dvcl.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | dvbssntr.j | . . . . 5 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
5 | dvbssntr.k | . . . . 5 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
6 | 4, 5 | dvfval 25343 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝐽)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ))) |
7 | 1, 2, 3, 6 | syl3anc 1371 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝐽)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ))) |
8 | dmss 5894 | . . 3 ⊢ ((𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ) → dom (𝑆 D 𝐹) ⊆ dom (((int‘𝐽)‘𝐴) × ℂ)) | |
9 | 7, 8 | simpl2im 504 | . 2 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ dom (((int‘𝐽)‘𝐴) × ℂ)) |
10 | dmxpss 6159 | . 2 ⊢ dom (((int‘𝐽)‘𝐴) × ℂ) ⊆ ((int‘𝐽)‘𝐴) | |
11 | 9, 10 | sstrdi 3990 | 1 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∖ cdif 3941 ⊆ wss 3944 {csn 4622 ∪ ciun 4990 ↦ cmpt 5224 × cxp 5667 dom cdm 5669 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 − cmin 11426 / cdiv 11853 ↾t crest 17348 TopOpenctopn 17349 ℂfldccnfld 20878 intcnt 22450 limℂ climc 25308 D cdv 25309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-pm 8806 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fi 9388 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-fz 13467 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-struct 17062 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17350 df-topn 17351 df-topgen 17371 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cnp 22661 df-xms 23755 df-ms 23756 df-limc 25312 df-dv 25313 |
This theorem is referenced by: dvbss 25347 dvnres 25377 dvcmulf 25391 dvcjbr 25395 dvmptcmul 25410 dvcnvre 25465 ftc1cn 25489 taylthlem1 25814 taylthlem2 25815 ulmdvlem3 25843 ftc1cnnc 36362 |
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