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Mirrors > Home > MPE Home > Th. List > dvmptres | Structured version Visualization version GIF version |
Description: Function-builder for derivative: restrict a derivative to an open subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvmptadd.s | β’ (π β π β {β, β}) |
dvmptadd.a | β’ ((π β§ π₯ β π) β π΄ β β) |
dvmptadd.b | β’ ((π β§ π₯ β π) β π΅ β π) |
dvmptadd.da | β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) |
dvmptres.y | β’ (π β π β π) |
dvmptres.j | β’ π½ = (πΎ βΎt π) |
dvmptres.k | β’ πΎ = (TopOpenββfld) |
dvmptres.t | β’ (π β π β π½) |
Ref | Expression |
---|---|
dvmptres | β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptadd.s | . 2 β’ (π β π β {β, β}) | |
2 | dvmptadd.a | . 2 β’ ((π β§ π₯ β π) β π΄ β β) | |
3 | dvmptadd.b | . 2 β’ ((π β§ π₯ β π) β π΅ β π) | |
4 | dvmptadd.da | . 2 β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) | |
5 | dvmptres.y | . 2 β’ (π β π β π) | |
6 | dvmptres.j | . 2 β’ π½ = (πΎ βΎt π) | |
7 | dvmptres.k | . 2 β’ πΎ = (TopOpenββfld) | |
8 | 7 | cnfldtop 24300 | . . . . 5 β’ πΎ β Top |
9 | resttop 22664 | . . . . 5 β’ ((πΎ β Top β§ π β {β, β}) β (πΎ βΎt π) β Top) | |
10 | 8, 1, 9 | sylancr 588 | . . . 4 β’ (π β (πΎ βΎt π) β Top) |
11 | 6, 10 | eqeltrid 2838 | . . 3 β’ (π β π½ β Top) |
12 | dvmptres.t | . . 3 β’ (π β π β π½) | |
13 | isopn3i 22586 | . . 3 β’ ((π½ β Top β§ π β π½) β ((intβπ½)βπ) = π) | |
14 | 11, 12, 13 | syl2anc 585 | . 2 β’ (π β ((intβπ½)βπ) = π) |
15 | 1, 2, 3, 4, 5, 6, 7, 14 | dvmptres2 25479 | 1 β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3949 {cpr 4631 β¦ cmpt 5232 βcfv 6544 (class class class)co 7409 βcc 11108 βcr 11109 βΎt crest 17366 TopOpenctopn 17367 βfldccnfld 20944 Topctop 22395 intcnt 22521 D cdv 25380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-cnp 22732 df-xms 23826 df-ms 23827 df-limc 25383 df-dv 25384 |
This theorem is referenced by: dvmptfsum 25492 dvexp3 25495 dvlipcn 25511 dvivthlem1 25525 lhop2 25532 dvfsumle 25538 dvfsumabs 25540 dvfsumlem2 25544 taylthlem2 25886 pserdvlem2 25940 advlog 26162 advlogexp 26163 logtayl 26168 loglesqrt 26266 dvatan 26440 log2sumbnd 27047 gg-dvfsumle 35182 gg-dvfsumlem2 35183 dvtan 36538 dvasin 36572 dvacos 36573 areacirclem1 36576 aks4d1p1p6 40938 dvmptconst 44631 dvmptidg 44633 itgsin0pilem1 44666 itgsbtaddcnst 44698 fourierdlem56 44878 fourierdlem60 44882 fourierdlem61 44883 fourierdlem62 44884 |
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