Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dvmptntr | Structured version Visualization version Unicode 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 | |
dvmptntr.x | |
dvmptntr.a | |
dvmptntr.j | ↾t |
dvmptntr.k | ℂfld |
dvmptntr.i |
Ref | Expression |
---|---|
dvmptntr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptntr.j | . . . . . . . . 9 ↾t | |
2 | dvmptntr.k | . . . . . . . . . . 11 ℂfld | |
3 | 2 | cnfldtopon 22586 | . . . . . . . . . 10 TopOn |
4 | dvmptntr.s | . . . . . . . . . 10 | |
5 | resttopon 20965 | . . . . . . . . . 10 TopOn ↾t TopOn | |
6 | 3, 4, 5 | sylancr 695 | . . . . . . . . 9 ↾t TopOn |
7 | 1, 6 | syl5eqel 2705 | . . . . . . . 8 TopOn |
8 | topontop 20718 | . . . . . . . 8 TopOn | |
9 | 7, 8 | syl 17 | . . . . . . 7 |
10 | dvmptntr.x | . . . . . . . 8 | |
11 | toponuni 20719 | . . . . . . . . 9 TopOn | |
12 | 7, 11 | syl 17 | . . . . . . . 8 |
13 | 10, 12 | sseqtrd 3641 | . . . . . . 7 |
14 | eqid 2622 | . . . . . . . 8 | |
15 | 14 | ntridm 20872 | . . . . . . 7 |
16 | 9, 13, 15 | syl2anc 693 | . . . . . 6 |
17 | dvmptntr.i | . . . . . . 7 | |
18 | 17 | fveq2d 6195 | . . . . . 6 |
19 | 16, 18 | eqtr3d 2658 | . . . . 5 |
20 | 19 | reseq2d 5396 | . . . 4 |
21 | dvmptntr.a | . . . . . 6 | |
22 | eqid 2622 | . . . . . 6 | |
23 | 21, 22 | fmptd 6385 | . . . . 5 |
24 | 2, 1 | dvres 23675 | . . . . 5 |
25 | 4, 23, 10, 10, 24 | syl22anc 1327 | . . . 4 |
26 | 14 | ntrss2 20861 | . . . . . . . 8 |
27 | 9, 13, 26 | syl2anc 693 | . . . . . . 7 |
28 | 17, 27 | eqsstr3d 3640 | . . . . . 6 |
29 | 28, 10 | sstrd 3613 | . . . . 5 |
30 | 2, 1 | dvres 23675 | . . . . 5 |
31 | 4, 23, 10, 29, 30 | syl22anc 1327 | . . . 4 |
32 | 20, 25, 31 | 3eqtr4d 2666 | . . 3 |
33 | ssid 3624 | . . . . 5 | |
34 | resmpt 5449 | . . . . 5 | |
35 | 33, 34 | mp1i 13 | . . . 4 |
36 | 35 | oveq2d 6666 | . . 3 |
37 | 32, 36 | eqtr3d 2658 | . 2 |
38 | 28 | resmptd 5452 | . . 3 |
39 | 38 | oveq2d 6666 | . 2 |
40 | 37, 39 | eqtr3d 2658 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wss 3574 cuni 4436 cmpt 4729 cres 5116 wf 5884 cfv 5888 (class class class)co 6650 cc 9934 ↾t crest 16081 ctopn 16082 ℂfldccnfld 19746 ctop 20698 TopOnctopon 20715 cnt 20821 cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-cnp 21032 df-xms 22125 df-ms 22126 df-limc 23630 df-dv 23631 |
This theorem is referenced by: rolle 23753 cmvth 23754 dvlip 23756 dvlipcn 23757 dvle 23770 dvfsumabs 23786 ftc2 23807 itgparts 23810 itgsubstlem 23811 lgamgulmlem2 24756 ftc2nc 33494 areacirc 33505 itgsin0pilem1 40165 itgsbtaddcnst 40198 |
Copyright terms: Public domain | W3C validator |