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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 | β’ (π β π β β) |
dvcl.f | β’ (π β πΉ:π΄βΆβ) |
dvcl.a | β’ (π β π΄ β π) |
dvbssntr.j | β’ π½ = (πΎ βΎt π) |
dvbssntr.k | β’ πΎ = (TopOpenββfld) |
Ref | Expression |
---|---|
dvbssntr | β’ (π β dom (π D πΉ) β ((intβπ½)βπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcl.s | . . . 4 β’ (π β π β β) | |
2 | dvcl.f | . . . 4 β’ (π β πΉ:π΄βΆβ) | |
3 | dvcl.a | . . . 4 β’ (π β π΄ β π) | |
4 | dvbssntr.j | . . . . 5 β’ π½ = (πΎ βΎt π) | |
5 | dvbssntr.k | . . . . 5 β’ πΎ = (TopOpenββfld) | |
6 | 4, 5 | dvfval 25747 | . . . 4 β’ ((π β β β§ πΉ:π΄βΆβ β§ π΄ β π) β ((π D πΉ) = βͺ π₯ β ((intβπ½)βπ΄)({π₯} Γ ((π§ β (π΄ β {π₯}) β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯)) β§ (π D πΉ) β (((intβπ½)βπ΄) Γ β))) |
7 | 1, 2, 3, 6 | syl3anc 1370 | . . 3 β’ (π β ((π D πΉ) = βͺ π₯ β ((intβπ½)βπ΄)({π₯} Γ ((π§ β (π΄ β {π₯}) β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯)) β§ (π D πΉ) β (((intβπ½)βπ΄) Γ β))) |
8 | dmss 5902 | . . 3 β’ ((π D πΉ) β (((intβπ½)βπ΄) Γ β) β dom (π D πΉ) β dom (((intβπ½)βπ΄) Γ β)) | |
9 | 7, 8 | simpl2im 503 | . 2 β’ (π β dom (π D πΉ) β dom (((intβπ½)βπ΄) Γ β)) |
10 | dmxpss 6170 | . 2 β’ dom (((intβπ½)βπ΄) Γ β) β ((intβπ½)βπ΄) | |
11 | 9, 10 | sstrdi 3994 | 1 β’ (π β dom (π D πΉ) β ((intβπ½)βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β cdif 3945 β wss 3948 {csn 4628 βͺ ciun 4997 β¦ cmpt 5231 Γ cxp 5674 dom cdm 5676 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcc 11114 β cmin 11451 / cdiv 11878 βΎt crest 17373 TopOpenctopn 17374 βfldccnfld 21234 intcnt 22842 limβ climc 25712 D cdv 25713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-topgen 17396 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-cnfld 21235 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cnp 23053 df-xms 24147 df-ms 24148 df-limc 25716 df-dv 25717 |
This theorem is referenced by: dvbss 25751 dvnres 25782 dvcmulf 25797 dvcjbr 25802 dvmptcmul 25817 dvcnvre 25873 ftc1cn 25899 taylthlem1 26225 taylthlem2 26226 ulmdvlem3 26254 gg-taylthlem2 35634 ftc1cnnc 37027 |
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