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Theorem ulmdv 25562

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
ulmdv	⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻)

Distinct variable groups: 𝑧,𝑘,𝐹 𝑧,𝐺 𝑧,𝐻 𝑘,𝑀 𝜑,𝑘,𝑧 𝑆,𝑘,𝑧 𝑘,𝑋,𝑧 𝑘,𝑍,𝑧 Allowed substitution hints: 𝐺(𝑘) 𝐻(𝑘) 𝑀(𝑧)

**Proof of Theorem ulmdv**
Step	Hyp	Ref	Expression
1		ulmdv.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		dvfg 25070	. . . . 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 25068	. . . . . . . 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 25560	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)
14		dvbsss 25066	. . . . . . . . . 10 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆
15	13, 14	eqsstrrdi 3976	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
16	15	ralrimiva 3103	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆)
17		uzid 12597	. . . . . . . . . 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 3562	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
21	5, 6, 20	dvbss 25065	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) ⊆ 𝑋)
22	8, 1, 9, 10, 6, 11, 12	ulmdvlem3 25561	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧))
23		vex 3436	. . . . . . . 8 ⊢ 𝑧 ∈ V
24		fvex 6787	. . . . . . . 8 ⊢ (𝐻‘𝑧) ∈ V
25	23, 24	breldm 5817	. . . . . . 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 3942	. . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋)
28	27	feq2d 6586	. . . 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 6601	. 2 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋)
31		ulmcl 25540	. . . 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 6601	. 2 ⊢ (𝜑 → 𝐻 Fn 𝑋)
34	3	ffund 6604	. . . 4 ⊢ (𝜑 → Fun (𝑆 D 𝐺))
35	34	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun (𝑆 D 𝐺))
36		funbrfv 6820	. . 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 6912	1 ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 {cpr 4563 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℂcc 10869 ℝcr 10870 ℤcz 12319 ℤ_≥cuz 12582 ⇝ cli 15193 D cdv 25027 ⇝_𝑢culm 25535

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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

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-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 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 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-ulm 25536

This theorem is referenced by: pserdvlem2 25587

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