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Mirrors > Home > MPE Home > Th. List > dvbss | Structured version Visualization version GIF version |
Description: The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-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) |
Ref | Expression |
---|---|
dvbss | ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcl.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
2 | dvcl.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | dvcl.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | eqid 2826 | . . 3 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆) | |
5 | eqid 2826 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
6 | 1, 2, 3, 4, 5 | dvbssntr 24064 | . 2 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝐴)) |
7 | 5 | cnfldtop 22958 | . . . 4 ⊢ (TopOpen‘ℂfld) ∈ Top |
8 | cnex 10334 | . . . . 5 ⊢ ℂ ∈ V | |
9 | ssexg 5030 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V) | |
10 | 1, 8, 9 | sylancl 582 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
11 | resttop 21336 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top) | |
12 | 7, 10, 11 | sylancr 583 | . . 3 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top) |
13 | 5 | cnfldtopon 22957 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
14 | resttopon 21337 | . . . . . 6 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
15 | 13, 1, 14 | sylancr 583 | . . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
16 | toponuni 21090 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)) |
18 | 3, 17 | sseqtrd 3867 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝑆)) |
19 | eqid 2826 | . . . 4 ⊢ ∪ ((TopOpen‘ℂfld) ↾t 𝑆) = ∪ ((TopOpen‘ℂfld) ↾t 𝑆) | |
20 | 19 | ntrss2 21233 | . . 3 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ 𝐴 ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝑆)) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝐴) ⊆ 𝐴) |
21 | 12, 18, 20 | syl2anc 581 | . 2 ⊢ (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝐴) ⊆ 𝐴) |
22 | 6, 21 | sstrd 3838 | 1 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3415 ⊆ wss 3799 ∪ cuni 4659 dom cdm 5343 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 ↾t crest 16435 TopOpenctopn 16436 ℂfldccnfld 20107 Topctop 21069 TopOnctopon 21086 intcnt 21193 D cdv 24027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fi 8587 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-q 12073 df-rp 12114 df-xneg 12233 df-xadd 12234 df-xmul 12235 df-fz 12621 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-plusg 16319 df-mulr 16320 df-starv 16321 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-rest 16437 df-topn 16438 df-topgen 16458 df-psmet 20099 df-xmet 20100 df-met 20101 df-bl 20102 df-mopn 20103 df-cnfld 20108 df-top 21070 df-topon 21087 df-topsp 21109 df-bases 21122 df-ntr 21196 df-cnp 21404 df-xms 22496 df-ms 22497 df-limc 24030 df-dv 24031 |
This theorem is referenced by: dvbsss 24066 dvres3 24077 dvres3a 24078 dvidlem 24079 dvcnp 24082 dvnff 24086 dvnres 24094 cpnord 24098 dvmulbr 24102 dvaddf 24105 dvmulf 24106 dvcmul 24107 dvcobr 24109 dvcof 24111 dvcjbr 24112 dvrec 24118 dvcnv 24140 dvlipcn 24157 dvlip2 24158 lhop 24179 dvtaylp 24524 ulmdv 24557 pserdv 24583 unbdqndv1 33032 unbdqndv2 33035 knoppndv 33058 fourierdlem80 41198 fourierdlem94 41212 fourierdlem113 41231 |
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