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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 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
2 | dvcl.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | dvcl.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | dvbssntr.j | . . . . . 6 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
5 | dvbssntr.k | . . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
6 | 4, 5 | dvfval 23998 | . . . . 5 ⊢ ((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 1491 | . . . 4 ⊢ (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 | 7 | simprd 490 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ)) |
9 | dmss 5524 | . . 3 ⊢ ((𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ) → dom (𝑆 D 𝐹) ⊆ dom (((int‘𝐽)‘𝐴) × ℂ)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ dom (((int‘𝐽)‘𝐴) × ℂ)) |
11 | dmxpss 5780 | . 2 ⊢ dom (((int‘𝐽)‘𝐴) × ℂ) ⊆ ((int‘𝐽)‘𝐴) | |
12 | 10, 11 | syl6ss 3808 | 1 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∖ cdif 3764 ⊆ wss 3767 {csn 4366 ∪ ciun 4708 ↦ cmpt 4920 × cxp 5308 dom cdm 5310 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 ℂcc 10220 − cmin 10554 / cdiv 10974 ↾t crest 16392 TopOpenctopn 16393 ℂfldccnfld 20064 intcnt 21146 limℂ climc 23963 D cdv 23964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fi 8557 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-q 12030 df-rp 12071 df-xneg 12189 df-xadd 12190 df-xmul 12191 df-fz 12577 df-seq 13052 df-exp 13111 df-cj 14176 df-re 14177 df-im 14178 df-sqrt 14312 df-abs 14313 df-struct 16182 df-ndx 16183 df-slot 16184 df-base 16186 df-plusg 16276 df-mulr 16277 df-starv 16278 df-tset 16282 df-ple 16283 df-ds 16285 df-unif 16286 df-rest 16394 df-topn 16395 df-topgen 16415 df-psmet 20056 df-xmet 20057 df-met 20058 df-bl 20059 df-mopn 20060 df-cnfld 20065 df-top 21023 df-topon 21040 df-topsp 21062 df-bases 21075 df-cnp 21357 df-xms 22449 df-ms 22450 df-limc 23967 df-dv 23968 |
This theorem is referenced by: dvbss 24002 dvnres 24031 dvcmulf 24045 dvcjbr 24049 dvmptcmul 24064 dvcnvre 24119 ftc1cn 24143 taylthlem1 24464 taylthlem2 24465 ulmdvlem3 24493 ftc1cnnc 33963 |
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