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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 25779 | . . . . 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 25777 | . . . . . . . 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 26277 | . . . . . . . . . 10 β’ ((π β§ π β π) β dom (π D (πΉβπ)) = π) |
14 | dvbsss 25775 | . . . . . . . . . 10 β’ dom (π D (πΉβπ)) β π | |
15 | 13, 14 | eqsstrrdi 4030 | . . . . . . . . 9 β’ ((π β§ π β π) β π β π) |
16 | 15 | ralrimiva 3138 | . . . . . . . 8 β’ (π β βπ β π π β π) |
17 | uzid 12836 | . . . . . . . . . 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 2836 | . . . . . . . 8 β’ (π β π β π) |
20 | 7, 16, 19 | rspcdva 3605 | . . . . . . 7 β’ (π β π β π) |
21 | 5, 6, 20 | dvbss 25774 | . . . . . 6 β’ (π β dom (π D πΊ) β π) |
22 | 8, 1, 9, 10, 6, 11, 12 | ulmdvlem3 26278 | . . . . . . 7 β’ ((π β§ π§ β π) β π§(π D πΊ)(π»βπ§)) |
23 | vex 3470 | . . . . . . . 8 β’ π§ β V | |
24 | fvex 6895 | . . . . . . . 8 β’ (π»βπ§) β V | |
25 | 23, 24 | breldm 5899 | . . . . . . 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 3996 | . . . . 5 β’ (π β dom (π D πΊ) = π) |
28 | 27 | feq2d 6694 | . . . 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 6709 | . 2 β’ (π β (π D πΊ) Fn π) |
31 | ulmcl 26257 | . . . 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 6709 | . 2 β’ (π β π» Fn π) |
34 | 3 | ffund 6712 | . . . 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 6933 | . . 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 7026 | 1 β’ (π β (π D πΊ) = π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3941 {cpr 4623 class class class wbr 5139 β¦ cmpt 5222 dom cdm 5667 Fun wfun 6528 βΆwf 6530 βcfv 6534 (class class class)co 7402 βm cmap 8817 βcc 11105 βcr 11106 β€cz 12557 β€β₯cuz 12821 β cli 15430 D cdv 25736 βπ’culm 26252 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-ntr 22868 df-cls 22869 df-nei 22946 df-lp 22984 df-perf 22985 df-cn 23075 df-cnp 23076 df-haus 23163 df-cmp 23235 df-tx 23410 df-hmeo 23603 df-fil 23694 df-fm 23786 df-flim 23787 df-flf 23788 df-xms 24170 df-ms 24171 df-tms 24172 df-cncf 24742 df-limc 25739 df-dv 25740 df-ulm 26253 |
This theorem is referenced by: pserdvlem2 26305 |
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