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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 24489 | . . . . 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 24487 | . . . . . . . 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 264 | . . . . . . . 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 24975 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) |
14 | dvbsss 24485 | . . . . . . . . . 10 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆 | |
15 | 13, 14 | eqsstrrdi 4010 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆) |
16 | 15 | ralrimiva 3182 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆) |
17 | uzid 12245 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
18 | 9, 17 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
19 | 18, 8 | eleqtrrdi 2924 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
20 | 7, 16, 19 | rspcdva 3617 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
21 | 5, 6, 20 | dvbss 24484 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) ⊆ 𝑋) |
22 | 8, 1, 9, 10, 6, 11, 12 | ulmdvlem3 24976 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧)) |
23 | vex 3489 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
24 | fvex 6669 | . . . . . . . 8 ⊢ (𝐻‘𝑧) ∈ V | |
25 | 23, 24 | breldm 5763 | . . . . . . 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 3976 | . . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) |
28 | 27 | feq2d 6486 | . . . 4 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ)) |
29 | 3, 28 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ) |
30 | 29 | ffnd 6501 | . 2 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋) |
31 | ulmcl 24955 | . . . 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 6501 | . 2 ⊢ (𝜑 → 𝐻 Fn 𝑋) |
34 | 3 | ffund 6504 | . . . 4 ⊢ (𝜑 → Fun (𝑆 D 𝐺)) |
35 | 34 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun (𝑆 D 𝐺)) |
36 | funbrfv 6702 | . . 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 6791 | 1 ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3924 {cpr 4555 class class class wbr 5052 ↦ cmpt 5132 dom cdm 5541 Fun wfun 6335 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 ↑m cmap 8392 ℂcc 10521 ℝcr 10522 ℤcz 11968 ℤ≥cuz 12230 ⇝ cli 14826 D cdv 24446 ⇝𝑢culm 24950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-fi 8861 df-sup 8892 df-inf 8893 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ioo 12729 df-ico 12731 df-icc 12732 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-limsup 14813 df-clim 14830 df-rlim 14831 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-hom 16572 df-cco 16573 df-rest 16679 df-topn 16680 df-0g 16698 df-gsum 16699 df-topgen 16700 df-pt 16701 df-prds 16704 df-xrs 16758 df-qtop 16763 df-imas 16764 df-xps 16766 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-mulg 18208 df-cntz 18430 df-cmn 18891 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-fbas 20525 df-fg 20526 df-cnfld 20529 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-lp 21727 df-perf 21728 df-cn 21818 df-cnp 21819 df-haus 21906 df-cmp 21978 df-tx 22153 df-hmeo 22346 df-fil 22437 df-fm 22529 df-flim 22530 df-flf 22531 df-xms 22913 df-ms 22914 df-tms 22915 df-cncf 23469 df-limc 24449 df-dv 24450 df-ulm 24951 |
This theorem is referenced by: pserdvlem2 25002 |
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