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Mirrors > Home > MPE Home > Th. List > ulmdv | Structured version Visualization version GIF version |
Description: If 𝐹 is a sequence of differentiable functions on 𝑋 which converge pointwise to 𝐺, and the derivatives of 𝐹(𝑛) converge uniformly to 𝐻, then 𝐺 is differentiable with derivative 𝐻. (Contributed by Mario Carneiro, 27-Feb-2015.) |
Ref | Expression |
---|---|
ulmdv.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ulmdv.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
ulmdv.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
ulmdv.f | ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑋)) |
ulmdv.g | ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
ulmdv.l | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) |
ulmdv.u | ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) |
Ref | Expression |
---|---|
ulmdv | ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmdv.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvfg 25080 | . . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) |
4 | recnprss 25078 | . . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
5 | 1, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
6 | ulmdv.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) | |
7 | biidd 261 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝑋 ⊆ 𝑆 ↔ 𝑋 ⊆ 𝑆)) | |
8 | ulmdv.z | . . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
9 | ulmdv.m | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | ulmdv.f | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑋)) | |
11 | ulmdv.l | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) | |
12 | ulmdv.u | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) | |
13 | 8, 1, 9, 10, 6, 11, 12 | ulmdvlem2 25570 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) |
14 | dvbsss 25076 | . . . . . . . . . 10 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆 | |
15 | 13, 14 | eqsstrrdi 3975 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆) |
16 | 15 | ralrimiva 3108 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆) |
17 | uzid 12607 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
18 | 9, 17 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
19 | 18, 8 | eleqtrrdi 2850 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
20 | 7, 16, 19 | rspcdva 3561 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
21 | 5, 6, 20 | dvbss 25075 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) ⊆ 𝑋) |
22 | 8, 1, 9, 10, 6, 11, 12 | ulmdvlem3 25571 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧)) |
23 | vex 3433 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
24 | fvex 6779 | . . . . . . . 8 ⊢ (𝐻‘𝑧) ∈ V | |
25 | 23, 24 | breldm 5810 | . . . . . . 7 ⊢ (𝑧(𝑆 D 𝐺)(𝐻‘𝑧) → 𝑧 ∈ dom (𝑆 D 𝐺)) |
26 | 22, 25 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ dom (𝑆 D 𝐺)) |
27 | 21, 26 | eqelssd 3941 | . . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) |
28 | 27 | feq2d 6578 | . . . 4 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ)) |
29 | 3, 28 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ) |
30 | 29 | ffnd 6593 | . 2 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋) |
31 | ulmcl 25550 | . . . 4 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻 → 𝐻:𝑋⟶ℂ) | |
32 | 12, 31 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻:𝑋⟶ℂ) |
33 | 32 | ffnd 6593 | . 2 ⊢ (𝜑 → 𝐻 Fn 𝑋) |
34 | 3 | ffund 6596 | . . . 4 ⊢ (𝜑 → Fun (𝑆 D 𝐺)) |
35 | 34 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun (𝑆 D 𝐺)) |
36 | funbrfv 6812 | . . 3 ⊢ (Fun (𝑆 D 𝐺) → (𝑧(𝑆 D 𝐺)(𝐻‘𝑧) → ((𝑆 D 𝐺)‘𝑧) = (𝐻‘𝑧))) | |
37 | 35, 22, 36 | sylc 65 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝑆 D 𝐺)‘𝑧) = (𝐻‘𝑧)) |
38 | 30, 33, 37 | eqfnfvd 6904 | 1 ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3886 {cpr 4563 class class class wbr 5073 ↦ cmpt 5156 dom cdm 5584 Fun wfun 6420 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 ↑m cmap 8602 ℂcc 10879 ℝcr 10880 ℤcz 12329 ℤ≥cuz 12592 ⇝ cli 15203 D cdv 25037 ⇝𝑢culm 25545 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-fl 13522 df-seq 13732 df-exp 13793 df-hash 14055 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-limsup 15190 df-clim 15207 df-rlim 15208 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-mulg 18711 df-cntz 18933 df-cmn 19398 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-lp 22297 df-perf 22298 df-cn 22388 df-cnp 22389 df-haus 22476 df-cmp 22548 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-xms 23483 df-ms 23484 df-tms 23485 df-cncf 24051 df-limc 25040 df-dv 25041 df-ulm 25546 |
This theorem is referenced by: pserdvlem2 25597 |
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