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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 23476 | . . . . 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 23474 | . . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
5 | 1, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
6 | ulmdv.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) | |
7 | ulmdv.m | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | uzid 11578 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
9 | 7, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
10 | ulmdv.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
11 | 9, 10 | syl6eleqr 2699 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
12 | ulmdv.f | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑋)) | |
13 | ulmdv.l | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) | |
14 | ulmdv.u | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) | |
15 | 10, 1, 7, 12, 6, 13, 14 | ulmdvlem2 23959 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) |
16 | dvbsss 23472 | . . . . . . . . . 10 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆 | |
17 | 15, 16 | syl6eqssr 3619 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆) |
18 | 17 | ralrimiva 2949 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆) |
19 | biidd 251 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑋 ⊆ 𝑆 ↔ 𝑋 ⊆ 𝑆)) | |
20 | 19 | rspcv 3278 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆 → 𝑋 ⊆ 𝑆)) |
21 | 11, 18, 20 | sylc 63 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
22 | 5, 6, 21 | dvbss 23471 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) ⊆ 𝑋) |
23 | 10, 1, 7, 12, 6, 13, 14 | ulmdvlem3 23960 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧)) |
24 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑧 ∈ V | |
25 | fvex 6113 | . . . . . . . . . 10 ⊢ (𝐻‘𝑧) ∈ V | |
26 | 24, 25 | breldm 5251 | . . . . . . . . 9 ⊢ (𝑧(𝑆 D 𝐺)(𝐻‘𝑧) → 𝑧 ∈ dom (𝑆 D 𝐺)) |
27 | 23, 26 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ dom (𝑆 D 𝐺)) |
28 | 27 | ex 449 | . . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝑋 → 𝑧 ∈ dom (𝑆 D 𝐺))) |
29 | 28 | ssrdv 3574 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ dom (𝑆 D 𝐺)) |
30 | 22, 29 | eqssd 3585 | . . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) |
31 | 30 | feq2d 5944 | . . . 4 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ)) |
32 | 3, 31 | mpbid 221 | . . 3 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ) |
33 | ffn 5958 | . . 3 ⊢ ((𝑆 D 𝐺):𝑋⟶ℂ → (𝑆 D 𝐺) Fn 𝑋) | |
34 | 32, 33 | syl 17 | . 2 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋) |
35 | ulmcl 23939 | . . . 4 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻 → 𝐻:𝑋⟶ℂ) | |
36 | 14, 35 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻:𝑋⟶ℂ) |
37 | ffn 5958 | . . 3 ⊢ (𝐻:𝑋⟶ℂ → 𝐻 Fn 𝑋) | |
38 | 36, 37 | syl 17 | . 2 ⊢ (𝜑 → 𝐻 Fn 𝑋) |
39 | ffun 5961 | . . . . 5 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺)) | |
40 | 3, 39 | syl 17 | . . . 4 ⊢ (𝜑 → Fun (𝑆 D 𝐺)) |
41 | 40 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun (𝑆 D 𝐺)) |
42 | funbrfv 6144 | . . 3 ⊢ (Fun (𝑆 D 𝐺) → (𝑧(𝑆 D 𝐺)(𝐻‘𝑧) → ((𝑆 D 𝐺)‘𝑧) = (𝐻‘𝑧))) | |
43 | 41, 23, 42 | sylc 63 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝑆 D 𝐺)‘𝑧) = (𝐻‘𝑧)) |
44 | 34, 38, 43 | eqfnfvd 6222 | 1 ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℂcc 9813 ℝcr 9814 ℤcz 11254 ℤ≥cuz 11563 ⇝ cli 14063 D cdv 23433 ⇝𝑢culm 23934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-ulm 23935 |
This theorem is referenced by: pserdvlem2 23986 |
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