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Mirrors > Home > MPE Home > Th. List > dvmptntr | Structured version Visualization version GIF version |
Description: Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvmptntr.s | β’ (π β π β β) |
dvmptntr.x | β’ (π β π β π) |
dvmptntr.a | β’ ((π β§ π₯ β π) β π΄ β β) |
dvmptntr.j | β’ π½ = (πΎ βΎt π) |
dvmptntr.k | β’ πΎ = (TopOpenββfld) |
dvmptntr.i | β’ (π β ((intβπ½)βπ) = π) |
Ref | Expression |
---|---|
dvmptntr | β’ (π β (π D (π₯ β π β¦ π΄)) = (π D (π₯ β π β¦ π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptntr.j | . . . . . . . . 9 β’ π½ = (πΎ βΎt π) | |
2 | dvmptntr.k | . . . . . . . . . . 11 β’ πΎ = (TopOpenββfld) | |
3 | 2 | cnfldtopon 24654 | . . . . . . . . . 10 β’ πΎ β (TopOnββ) |
4 | dvmptntr.s | . . . . . . . . . 10 β’ (π β π β β) | |
5 | resttopon 23020 | . . . . . . . . . 10 β’ ((πΎ β (TopOnββ) β§ π β β) β (πΎ βΎt π) β (TopOnβπ)) | |
6 | 3, 4, 5 | sylancr 586 | . . . . . . . . 9 β’ (π β (πΎ βΎt π) β (TopOnβπ)) |
7 | 1, 6 | eqeltrid 2831 | . . . . . . . 8 β’ (π β π½ β (TopOnβπ)) |
8 | topontop 22770 | . . . . . . . 8 β’ (π½ β (TopOnβπ) β π½ β Top) | |
9 | 7, 8 | syl 17 | . . . . . . 7 β’ (π β π½ β Top) |
10 | dvmptntr.x | . . . . . . . 8 β’ (π β π β π) | |
11 | toponuni 22771 | . . . . . . . . 9 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 β’ (π β π = βͺ π½) |
13 | 10, 12 | sseqtrd 4017 | . . . . . . 7 β’ (π β π β βͺ π½) |
14 | eqid 2726 | . . . . . . . 8 β’ βͺ π½ = βͺ π½ | |
15 | 14 | ntridm 22927 | . . . . . . 7 β’ ((π½ β Top β§ π β βͺ π½) β ((intβπ½)β((intβπ½)βπ)) = ((intβπ½)βπ)) |
16 | 9, 13, 15 | syl2anc 583 | . . . . . 6 β’ (π β ((intβπ½)β((intβπ½)βπ)) = ((intβπ½)βπ)) |
17 | dvmptntr.i | . . . . . . 7 β’ (π β ((intβπ½)βπ) = π) | |
18 | 17 | fveq2d 6889 | . . . . . 6 β’ (π β ((intβπ½)β((intβπ½)βπ)) = ((intβπ½)βπ)) |
19 | 16, 18 | eqtr3d 2768 | . . . . 5 β’ (π β ((intβπ½)βπ) = ((intβπ½)βπ)) |
20 | 19 | reseq2d 5975 | . . . 4 β’ (π β ((π D (π₯ β π β¦ π΄)) βΎ ((intβπ½)βπ)) = ((π D (π₯ β π β¦ π΄)) βΎ ((intβπ½)βπ))) |
21 | dvmptntr.a | . . . . . 6 β’ ((π β§ π₯ β π) β π΄ β β) | |
22 | 21 | fmpttd 7110 | . . . . 5 β’ (π β (π₯ β π β¦ π΄):πβΆβ) |
23 | 2, 1 | dvres 25795 | . . . . 5 β’ (((π β β β§ (π₯ β π β¦ π΄):πβΆβ) β§ (π β π β§ π β π)) β (π D ((π₯ β π β¦ π΄) βΎ π)) = ((π D (π₯ β π β¦ π΄)) βΎ ((intβπ½)βπ))) |
24 | 4, 22, 10, 10, 23 | syl22anc 836 | . . . 4 β’ (π β (π D ((π₯ β π β¦ π΄) βΎ π)) = ((π D (π₯ β π β¦ π΄)) βΎ ((intβπ½)βπ))) |
25 | 14 | ntrss2 22916 | . . . . . . . 8 β’ ((π½ β Top β§ π β βͺ π½) β ((intβπ½)βπ) β π) |
26 | 9, 13, 25 | syl2anc 583 | . . . . . . 7 β’ (π β ((intβπ½)βπ) β π) |
27 | 17, 26 | eqsstrrd 4016 | . . . . . 6 β’ (π β π β π) |
28 | 27, 10 | sstrd 3987 | . . . . 5 β’ (π β π β π) |
29 | 2, 1 | dvres 25795 | . . . . 5 β’ (((π β β β§ (π₯ β π β¦ π΄):πβΆβ) β§ (π β π β§ π β π)) β (π D ((π₯ β π β¦ π΄) βΎ π)) = ((π D (π₯ β π β¦ π΄)) βΎ ((intβπ½)βπ))) |
30 | 4, 22, 10, 28, 29 | syl22anc 836 | . . . 4 β’ (π β (π D ((π₯ β π β¦ π΄) βΎ π)) = ((π D (π₯ β π β¦ π΄)) βΎ ((intβπ½)βπ))) |
31 | 20, 24, 30 | 3eqtr4d 2776 | . . 3 β’ (π β (π D ((π₯ β π β¦ π΄) βΎ π)) = (π D ((π₯ β π β¦ π΄) βΎ π))) |
32 | ssid 3999 | . . . . 5 β’ π β π | |
33 | resmpt 6031 | . . . . 5 β’ (π β π β ((π₯ β π β¦ π΄) βΎ π) = (π₯ β π β¦ π΄)) | |
34 | 32, 33 | mp1i 13 | . . . 4 β’ (π β ((π₯ β π β¦ π΄) βΎ π) = (π₯ β π β¦ π΄)) |
35 | 34 | oveq2d 7421 | . . 3 β’ (π β (π D ((π₯ β π β¦ π΄) βΎ π)) = (π D (π₯ β π β¦ π΄))) |
36 | 31, 35 | eqtr3d 2768 | . 2 β’ (π β (π D ((π₯ β π β¦ π΄) βΎ π)) = (π D (π₯ β π β¦ π΄))) |
37 | 27 | resmptd 6034 | . . 3 β’ (π β ((π₯ β π β¦ π΄) βΎ π) = (π₯ β π β¦ π΄)) |
38 | 37 | oveq2d 7421 | . 2 β’ (π β (π D ((π₯ β π β¦ π΄) βΎ π)) = (π D (π₯ β π β¦ π΄))) |
39 | 36, 38 | eqtr3d 2768 | 1 β’ (π β (π D (π₯ β π β¦ π΄)) = (π D (π₯ β π β¦ π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 βͺ cuni 4902 β¦ cmpt 5224 βΎ cres 5671 βΆwf 6533 βcfv 6537 (class class class)co 7405 βcc 11110 βΎt crest 17375 TopOpenctopn 17376 βfldccnfld 21240 Topctop 22750 TopOnctopon 22767 intcnt 22876 D cdv 25747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 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 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-fz 13491 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-rest 17377 df-topn 17378 df-topgen 17398 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-cnp 23087 df-xms 24181 df-ms 24182 df-limc 25750 df-dv 25751 |
This theorem is referenced by: rolle 25877 cmvth 25878 cmvthOLD 25879 dvlip 25881 dvlipcn 25882 dvle 25895 dvfsumabs 25912 ftc2 25934 itgparts 25937 itgsubstlem 25938 lgamgulmlem2 26917 ftc2nc 37083 areacirc 37094 itgsin0pilem1 45235 itgsbtaddcnst 45267 |
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