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Mirrors > Home > MPE Home > Th. List > ulmdv | Structured version Visualization version GIF version |
Description: If πΉ is a sequence of differentiable functions on π which converge pointwise to πΊ, and the derivatives of πΉ(π) converge uniformly to π», then πΊ is differentiable with derivative π». (Contributed by Mario Carneiro, 27-Feb-2015.) |
Ref | Expression |
---|---|
ulmdv.z | β’ π = (β€β₯βπ) |
ulmdv.s | β’ (π β π β {β, β}) |
ulmdv.m | β’ (π β π β β€) |
ulmdv.f | β’ (π β πΉ:πβΆ(β βm π)) |
ulmdv.g | β’ (π β πΊ:πβΆβ) |
ulmdv.l | β’ ((π β§ π§ β π) β (π β π β¦ ((πΉβπ)βπ§)) β (πΊβπ§)) |
ulmdv.u | β’ (π β (π β π β¦ (π D (πΉβπ)))(βπ’βπ)π») |
Ref | Expression |
---|---|
ulmdv | β’ (π β (π D πΊ) = π») |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmdv.s | . . . . 5 β’ (π β π β {β, β}) | |
2 | dvfg 25286 | . . . . 5 β’ (π β {β, β} β (π D πΊ):dom (π D πΊ)βΆβ) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β (π D πΊ):dom (π D πΊ)βΆβ) |
4 | recnprss 25284 | . . . . . . . 8 β’ (π β {β, β} β π β β) | |
5 | 1, 4 | syl 17 | . . . . . . 7 β’ (π β π β β) |
6 | ulmdv.g | . . . . . . 7 β’ (π β πΊ:πβΆβ) | |
7 | biidd 262 | . . . . . . . 8 β’ (π = π β (π β π β π β π)) | |
8 | ulmdv.z | . . . . . . . . . . 11 β’ π = (β€β₯βπ) | |
9 | ulmdv.m | . . . . . . . . . . 11 β’ (π β π β β€) | |
10 | ulmdv.f | . . . . . . . . . . 11 β’ (π β πΉ:πβΆ(β βm π)) | |
11 | ulmdv.l | . . . . . . . . . . 11 β’ ((π β§ π§ β π) β (π β π β¦ ((πΉβπ)βπ§)) β (πΊβπ§)) | |
12 | ulmdv.u | . . . . . . . . . . 11 β’ (π β (π β π β¦ (π D (πΉβπ)))(βπ’βπ)π») | |
13 | 8, 1, 9, 10, 6, 11, 12 | ulmdvlem2 25776 | . . . . . . . . . 10 β’ ((π β§ π β π) β dom (π D (πΉβπ)) = π) |
14 | dvbsss 25282 | . . . . . . . . . 10 β’ dom (π D (πΉβπ)) β π | |
15 | 13, 14 | eqsstrrdi 4004 | . . . . . . . . 9 β’ ((π β§ π β π) β π β π) |
16 | 15 | ralrimiva 3144 | . . . . . . . 8 β’ (π β βπ β π π β π) |
17 | uzid 12785 | . . . . . . . . . 10 β’ (π β β€ β π β (β€β₯βπ)) | |
18 | 9, 17 | syl 17 | . . . . . . . . 9 β’ (π β π β (β€β₯βπ)) |
19 | 18, 8 | eleqtrrdi 2849 | . . . . . . . 8 β’ (π β π β π) |
20 | 7, 16, 19 | rspcdva 3585 | . . . . . . 7 β’ (π β π β π) |
21 | 5, 6, 20 | dvbss 25281 | . . . . . 6 β’ (π β dom (π D πΊ) β π) |
22 | 8, 1, 9, 10, 6, 11, 12 | ulmdvlem3 25777 | . . . . . . 7 β’ ((π β§ π§ β π) β π§(π D πΊ)(π»βπ§)) |
23 | vex 3452 | . . . . . . . 8 β’ π§ β V | |
24 | fvex 6860 | . . . . . . . 8 β’ (π»βπ§) β V | |
25 | 23, 24 | breldm 5869 | . . . . . . 7 β’ (π§(π D πΊ)(π»βπ§) β π§ β dom (π D πΊ)) |
26 | 22, 25 | syl 17 | . . . . . 6 β’ ((π β§ π§ β π) β π§ β dom (π D πΊ)) |
27 | 21, 26 | eqelssd 3970 | . . . . 5 β’ (π β dom (π D πΊ) = π) |
28 | 27 | feq2d 6659 | . . . 4 β’ (π β ((π D πΊ):dom (π D πΊ)βΆβ β (π D πΊ):πβΆβ)) |
29 | 3, 28 | mpbid 231 | . . 3 β’ (π β (π D πΊ):πβΆβ) |
30 | 29 | ffnd 6674 | . 2 β’ (π β (π D πΊ) Fn π) |
31 | ulmcl 25756 | . . . 4 β’ ((π β π β¦ (π D (πΉβπ)))(βπ’βπ)π» β π»:πβΆβ) | |
32 | 12, 31 | syl 17 | . . 3 β’ (π β π»:πβΆβ) |
33 | 32 | ffnd 6674 | . 2 β’ (π β π» Fn π) |
34 | 3 | ffund 6677 | . . . 4 β’ (π β Fun (π D πΊ)) |
35 | 34 | adantr 482 | . . 3 β’ ((π β§ π§ β π) β Fun (π D πΊ)) |
36 | funbrfv 6898 | . . 3 β’ (Fun (π D πΊ) β (π§(π D πΊ)(π»βπ§) β ((π D πΊ)βπ§) = (π»βπ§))) | |
37 | 35, 22, 36 | sylc 65 | . 2 β’ ((π β§ π§ β π) β ((π D πΊ)βπ§) = (π»βπ§)) |
38 | 30, 33, 37 | eqfnfvd 6990 | 1 β’ (π β (π D πΊ) = π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3915 {cpr 4593 class class class wbr 5110 β¦ cmpt 5193 dom cdm 5638 Fun wfun 6495 βΆwf 6497 βcfv 6501 (class class class)co 7362 βm cmap 8772 βcc 11056 βcr 11057 β€cz 12506 β€β₯cuz 12770 β cli 15373 D cdv 25243 βπ’culm 25751 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-mulg 18880 df-cntz 19104 df-cmn 19571 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-cmp 22754 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 df-ulm 25752 |
This theorem is referenced by: pserdvlem2 25803 |
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