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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 24307 | . . . . 5 β’ πΎ β Top |
9 | resttop 22671 | . . . . 5 β’ ((πΎ β Top β§ π β {β, β}) β (πΎ βΎt π) β Top) | |
10 | 8, 1, 9 | sylancr 587 | . . . 4 β’ (π β (πΎ βΎt π) β Top) |
11 | 6, 10 | eqeltrid 2837 | . . 3 β’ (π β π½ β Top) |
12 | dvmptres.t | . . 3 β’ (π β π β π½) | |
13 | isopn3i 22593 | . . 3 β’ ((π½ β Top β§ π β π½) β ((intβπ½)βπ) = π) | |
14 | 11, 12, 13 | syl2anc 584 | . 2 β’ (π β ((intβπ½)βπ) = π) |
15 | 1, 2, 3, 4, 5, 6, 7, 14 | dvmptres2 25486 | 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 396 = wceq 1541 β wcel 2106 β wss 3948 {cpr 4630 β¦ cmpt 5231 βcfv 6543 (class class class)co 7411 βcc 11110 βcr 11111 βΎt crest 17368 TopOpenctopn 17369 βfldccnfld 20950 Topctop 22402 intcnt 22528 D cdv 25387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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-iin 5000 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-fz 13487 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17082 df-slot 17117 df-ndx 17129 df-base 17147 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-rest 17370 df-topn 17371 df-topgen 17391 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-cnp 22739 df-xms 23833 df-ms 23834 df-limc 25390 df-dv 25391 |
This theorem is referenced by: dvmptfsum 25499 dvexp3 25502 dvlipcn 25518 dvivthlem1 25532 lhop2 25539 dvfsumle 25545 dvfsumabs 25547 dvfsumlem2 25551 taylthlem2 25893 pserdvlem2 25947 advlog 26169 advlogexp 26170 logtayl 26175 loglesqrt 26273 dvatan 26447 log2sumbnd 27054 gg-dvfsumle 35251 gg-dvfsumlem2 35252 dvtan 36624 dvasin 36658 dvacos 36659 areacirclem1 36662 aks4d1p1p6 41024 dvmptconst 44710 dvmptidg 44712 itgsin0pilem1 44745 itgsbtaddcnst 44777 fourierdlem56 44957 fourierdlem60 44961 fourierdlem61 44962 fourierdlem62 44963 |
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