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Theorem ulmdv 26386

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 25891	. . . . 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 25889	. . . . . . . 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 263	. . . . . . . 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 26384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)
14		dvbsss 25887	. . . . . . . . . 10 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆
15	13, 14	eqsstrrdi 3960	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
16	15	ralrimiva 3131	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆)
17		uzid 12794	. . . . . . . . . 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 25886	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) ⊆ 𝑋)
22	8, 1, 9, 10, 6, 11, 12	ulmdvlem3 26385	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧))
23		vex 3435	. . . . . . . 8 ⊢ 𝑧 ∈ V
24		fvex 6840	. . . . . . . 8 ⊢ (𝐻‘𝑧) ∈ V
25	23, 24	breldm 5850	. . . . . . 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 3936	. . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋)
28	27	feq2d 6639	. . . 4 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ))
29	3, 28	mpbid 233	. . 3 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ)
30	29	ffnd 6656	. 2 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋)
31		ulmcl 26364	. . . 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 6656	. 2 ⊢ (𝜑 → 𝐻 Fn 𝑋)
34	3	ffund 6659	. . . 4 ⊢ (𝜑 → Fun (𝑆 D 𝐺))
35	34	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun (𝑆 D 𝐺))
36		funbrfv 6875	. . 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 6974	1 ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ⊆ wss 3883 {cpr 4557 class class class wbr 5072 ↦ cmpt 5153 dom cdm 5618 Fun wfun 6479 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 ℂcc 11027 ℝcr 11028 ℤcz 12515 ℤ_≥cuz 12779 ⇝ cli 15437 D cdv 25848 ⇝_𝑢culm 26359

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852 df-ulm 26360

This theorem is referenced by: pserdvlem2 26411

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