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Theorem ulmdv 26432

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 25937	. . . . 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 25935	. . . . . . . 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 26430	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)
14		dvbsss 25933	. . . . . . . . . 10 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆
15	13, 14	eqsstrrdi 3972	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
16	15	ralrimiva 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆)
17		uzid 12840	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
18	9, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
19	18, 8	eleqtrrdi 2863	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
20	7, 16, 19	rspcdva 3573	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
21	5, 6, 20	dvbss 25932	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) ⊆ 𝑋)
22	8, 1, 9, 10, 6, 11, 12	ulmdvlem3 26431	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧))
23		vex 3448	. . . . . . . 8 ⊢ 𝑧 ∈ V
24		fvex 6865	. . . . . . . 8 ⊢ (𝐻‘𝑧) ∈ V
25	23, 24	breldm 5873	. . . . . . 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 3948	. . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋)
28	27	feq2d 6660	. . . 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 6677	. 2 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋)
31		ulmcl 26410	. . . 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 6677	. 2 ⊢ (𝜑 → 𝐻 Fn 𝑋)
34	3	ffund 6681	. . . 4 ⊢ (𝜑 → Fun (𝑆 D 𝐺))
35	34	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun (𝑆 D 𝐺))
36		funbrfv 6900	. . 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 6999	1 ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ⊆ wss 3895 {cpr 4574 class class class wbr 5090 ↦ cmpt 5171 dom cdm 5636 Fun wfun 6500 ⟶wf 6502 ‘cfv 6506 (class class class)co 7381 ↑_m cmap 8792 ℂcc 11057 ℝcr 11058 ℤcz 12554 ℤ_≥cuz 12825 ⇝ cli 15483 D cdv 25894 ⇝_𝑢culm 26405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-seq 14001 df-exp 14061 df-hash 14330 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-limsup 15470 df-clim 15487 df-rlim 15488 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-mulg 19082 df-cntz 19329 df-cmn 19794 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-fbas 21390 df-fg 21391 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24349 df-ms 24350 df-tms 24351 df-cncf 24909 df-limc 25897 df-dv 25898 df-ulm 26406

This theorem is referenced by: pserdvlem2 26457

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