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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 25834 | . . . . 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 25832 | . . . . . . . 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 26336 | . . . . . . . . . 10 β’ ((π β§ π β π) β dom (π D (πΉβπ)) = π) |
14 | dvbsss 25830 | . . . . . . . . . 10 β’ dom (π D (πΉβπ)) β π | |
15 | 13, 14 | eqsstrrdi 4035 | . . . . . . . . 9 β’ ((π β§ π β π) β π β π) |
16 | 15 | ralrimiva 3143 | . . . . . . . 8 β’ (π β βπ β π π β π) |
17 | uzid 12867 | . . . . . . . . . 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 2840 | . . . . . . . 8 β’ (π β π β π) |
20 | 7, 16, 19 | rspcdva 3610 | . . . . . . 7 β’ (π β π β π) |
21 | 5, 6, 20 | dvbss 25829 | . . . . . 6 β’ (π β dom (π D πΊ) β π) |
22 | 8, 1, 9, 10, 6, 11, 12 | ulmdvlem3 26337 | . . . . . . 7 β’ ((π β§ π§ β π) β π§(π D πΊ)(π»βπ§)) |
23 | vex 3475 | . . . . . . . 8 β’ π§ β V | |
24 | fvex 6910 | . . . . . . . 8 β’ (π»βπ§) β V | |
25 | 23, 24 | breldm 5911 | . . . . . . 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 4001 | . . . . 5 β’ (π β dom (π D πΊ) = π) |
28 | 27 | feq2d 6708 | . . . 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 6723 | . 2 β’ (π β (π D πΊ) Fn π) |
31 | ulmcl 26316 | . . . 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 6723 | . 2 β’ (π β π» Fn π) |
34 | 3 | ffund 6726 | . . . 4 β’ (π β Fun (π D πΊ)) |
35 | 34 | adantr 480 | . . 3 β’ ((π β§ π§ β π) β Fun (π D πΊ)) |
36 | funbrfv 6948 | . . 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 7043 | 1 β’ (π β (π D πΊ) = π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3947 {cpr 4631 class class class wbr 5148 β¦ cmpt 5231 dom cdm 5678 Fun wfun 6542 βΆwf 6544 βcfv 6548 (class class class)co 7420 βm cmap 8844 βcc 11136 βcr 11137 β€cz 12588 β€β₯cuz 12852 β cli 15460 D cdv 25791 βπ’culm 26311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-cmp 23290 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24225 df-ms 24226 df-tms 24227 df-cncf 24797 df-limc 25794 df-dv 25795 df-ulm 26312 |
This theorem is referenced by: pserdvlem2 26364 |
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