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Theorem ulmdv 26368

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 25873	. . . . 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 25871	. . . . . . . 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 262	. . . . . . . 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 26366	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)
14		dvbsss 25869	. . . . . . . . . 10 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆
15	13, 14	eqsstrrdi 3967	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
16	15	ralrimiva 3129	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆)
17		uzid 12803	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
18	9, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
19	18, 8	eleqtrrdi 2847	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
20	7, 16, 19	rspcdva 3565	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
21	5, 6, 20	dvbss 25868	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐺) ⊆ 𝑋)
22	8, 1, 9, 10, 6, 11, 12	ulmdvlem3 26367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧))
23		vex 3433	. . . . . . . 8 ⊢ 𝑧 ∈ V
24		fvex 6853	. . . . . . . 8 ⊢ (𝐻‘𝑧) ∈ V
25	23, 24	breldm 5863	. . . . . . 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 3943	. . . . 5 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋)
28	27	feq2d 6652	. . . 4 ⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ))
29	3, 28	mpbid 232	. . 3 ⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ)
30	29	ffnd 6669	. 2 ⊢ (𝜑 → (𝑆 D 𝐺) Fn 𝑋)
31		ulmcl 26346	. . . 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 6669	. 2 ⊢ (𝜑 → 𝐻 Fn 𝑋)
34	3	ffund 6672	. . . 4 ⊢ (𝜑 → Fun (𝑆 D 𝐺))
35	34	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun (𝑆 D 𝐺))
36		funbrfv 6888	. . 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 6986	1 ⊢ (𝜑 → (𝑆 D 𝐺) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⊆ wss 3889 {cpr 4569 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 Fun wfun 6492 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 ℂcc 11036 ℝcr 11037 ℤcz 12524 ℤ_≥cuz 12788 ⇝ cli 15446 D cdv 25830 ⇝_𝑢culm 26341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-ulm 26342

This theorem is referenced by: pserdvlem2 26393

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