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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 24521 | . . . . 5 β’ πΎ β Top |
9 | resttop 22885 | . . . . 5 β’ ((πΎ β Top β§ π β {β, β}) β (πΎ βΎt π) β Top) | |
10 | 8, 1, 9 | sylancr 586 | . . . 4 β’ (π β (πΎ βΎt π) β Top) |
11 | 6, 10 | eqeltrid 2836 | . . 3 β’ (π β π½ β Top) |
12 | dvmptres.t | . . 3 β’ (π β π β π½) | |
13 | isopn3i 22807 | . . 3 β’ ((π½ β Top β§ π β π½) β ((intβπ½)βπ) = π) | |
14 | 11, 12, 13 | syl2anc 583 | . 2 β’ (π β ((intβπ½)βπ) = π) |
15 | 1, 2, 3, 4, 5, 6, 7, 14 | dvmptres2 25712 | 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 395 = wceq 1540 β wcel 2105 β wss 3949 {cpr 4631 β¦ cmpt 5232 βcfv 6544 (class class class)co 7412 βcc 11111 βcr 11112 βΎt crest 17371 TopOpenctopn 17372 βfldccnfld 21145 Topctop 22616 intcnt 22742 D cdv 25613 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 |
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 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fi 9409 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-fz 13490 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-rest 17373 df-topn 17374 df-topgen 17394 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-cnp 22953 df-xms 24047 df-ms 24048 df-limc 25616 df-dv 25617 |
This theorem is referenced by: dvmptfsum 25725 dvexp3 25728 dvlipcn 25744 dvivthlem1 25758 lhop2 25765 dvfsumle 25771 dvfsumabs 25773 dvfsumlem2 25777 taylthlem2 26119 pserdvlem2 26173 advlog 26395 advlogexp 26396 logtayl 26401 loglesqrt 26499 dvatan 26673 log2sumbnd 27280 gg-dvfsumle 35469 gg-dvfsumlem2 35470 dvtan 36842 dvasin 36876 dvacos 36877 areacirclem1 36880 aks4d1p1p6 41245 dvmptconst 44931 dvmptidg 44933 itgsin0pilem1 44966 itgsbtaddcnst 44998 fourierdlem56 45178 fourierdlem60 45182 fourierdlem61 45183 fourierdlem62 45184 |
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