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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 | β’ (π β π β β) |
dvcl.f | β’ (π β πΉ:π΄βΆβ) |
dvcl.a | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
dvbss | β’ (π β dom (π D πΉ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcl.s | . . 3 β’ (π β π β β) | |
2 | dvcl.f | . . 3 β’ (π β πΉ:π΄βΆβ) | |
3 | dvcl.a | . . 3 β’ (π β π΄ β π) | |
4 | eqid 2728 | . . 3 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
5 | eqid 2728 | . . 3 β’ (TopOpenββfld) = (TopOpenββfld) | |
6 | 1, 2, 3, 4, 5 | dvbssntr 25828 | . 2 β’ (π β dom (π D πΉ) β ((intβ((TopOpenββfld) βΎt π))βπ΄)) |
7 | 5 | cnfldtop 24699 | . . . 4 β’ (TopOpenββfld) β Top |
8 | cnex 11219 | . . . . 5 β’ β β V | |
9 | ssexg 5323 | . . . . 5 β’ ((π β β β§ β β V) β π β V) | |
10 | 1, 8, 9 | sylancl 585 | . . . 4 β’ (π β π β V) |
11 | resttop 23063 | . . . 4 β’ (((TopOpenββfld) β Top β§ π β V) β ((TopOpenββfld) βΎt π) β Top) | |
12 | 7, 10, 11 | sylancr 586 | . . 3 β’ (π β ((TopOpenββfld) βΎt π) β Top) |
13 | 5 | cnfldtopon 24698 | . . . . . 6 β’ (TopOpenββfld) β (TopOnββ) |
14 | resttopon 23064 | . . . . . 6 β’ (((TopOpenββfld) β (TopOnββ) β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) | |
15 | 13, 1, 14 | sylancr 586 | . . . . 5 β’ (π β ((TopOpenββfld) βΎt π) β (TopOnβπ)) |
16 | toponuni 22815 | . . . . 5 β’ (((TopOpenββfld) βΎt π) β (TopOnβπ) β π = βͺ ((TopOpenββfld) βΎt π)) | |
17 | 15, 16 | syl 17 | . . . 4 β’ (π β π = βͺ ((TopOpenββfld) βΎt π)) |
18 | 3, 17 | sseqtrd 4020 | . . 3 β’ (π β π΄ β βͺ ((TopOpenββfld) βΎt π)) |
19 | eqid 2728 | . . . 4 β’ βͺ ((TopOpenββfld) βΎt π) = βͺ ((TopOpenββfld) βΎt π) | |
20 | 19 | ntrss2 22960 | . . 3 β’ ((((TopOpenββfld) βΎt π) β Top β§ π΄ β βͺ ((TopOpenββfld) βΎt π)) β ((intβ((TopOpenββfld) βΎt π))βπ΄) β π΄) |
21 | 12, 18, 20 | syl2anc 583 | . 2 β’ (π β ((intβ((TopOpenββfld) βΎt π))βπ΄) β π΄) |
22 | 6, 21 | sstrd 3990 | 1 β’ (π β dom (π D πΉ) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3471 β wss 3947 βͺ cuni 4908 dom cdm 5678 βΆwf 6544 βcfv 6548 (class class class)co 7420 βcc 11136 βΎt crest 17401 TopOpenctopn 17402 βfldccnfld 21278 Topctop 22794 TopOnctopon 22811 intcnt 22920 D cdv 25791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fi 9434 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-rest 17403 df-topn 17404 df-topgen 17424 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-cnfld 21279 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-ntr 22923 df-cnp 23131 df-xms 24225 df-ms 24226 df-limc 25794 df-dv 25795 |
This theorem is referenced by: dvbsss 25830 dvres3 25841 dvres3a 25842 dvidlem 25843 dvcnp 25847 dvnff 25852 dvnres 25860 cpnord 25864 dvmulbr 25868 dvmulbrOLD 25869 dvaddf 25872 dvmulf 25873 dvcmul 25874 dvcobr 25876 dvcobrOLD 25877 dvcof 25879 dvcjbr 25880 dvrec 25886 dvcnv 25908 dvlipcn 25926 dvlip2 25927 lhop 25948 dvtaylp 26304 ulmdv 26338 pserdv 26365 unbdqndv1 35983 unbdqndv2 35986 knoppndv 36009 fourierdlem80 45574 fourierdlem94 45588 fourierdlem113 45607 |
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