etransclem34 - Mathbox for Glauco Siliprandi

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Theorem etransclem34 47002

Description: The 𝑁-th derivative of 𝐹 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem34.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem34.a	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem34.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem34.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem34.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑘 ∈ (1...𝑀)((𝑥 − 𝑘)↑𝑃)))
etransclem34.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
etransclem34.h	⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
etransclem34.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑛})

**Assertion**
Ref	Expression
etransclem34	⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (𝑋–cn→ℂ))

Distinct variable groups: 𝐶,𝑐,𝑘,𝑥 𝐹,𝑐 𝐻,𝑐,𝑘,𝑛,𝑥 𝑀,𝑐,𝑘,𝑥,𝑛 𝑁,𝑐,𝑘,𝑥,𝑛 𝑃,𝑘,𝑥 𝑆,𝑐,𝑘,𝑛,𝑥 𝑋,𝑐,𝑘,𝑥,𝑛 𝜑,𝑐,𝑘,𝑥,𝑛 Allowed substitution hints: 𝐶(𝑛) 𝑃(𝑛,𝑐) 𝐹(𝑥,𝑘,𝑛)

Proof of Theorem etransclem34 Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem34.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		etransclem34.a	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
3		etransclem34.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4		etransclem34.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
5		etransclem34.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑘 ∈ (1...𝑀)((𝑥 − 𝑘)↑𝑃)))
6		etransclem34.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		etransclem34.h	. . 3 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
8		etransclem34.c	. . 3 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑛})
9	1, 2, 3, 4, 5, 6, 7, 8	etransclem30 46998	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) · ∏𝑘 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))))
10	1, 2	dvdmsscn 46670	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
11	8, 6	etransclem16 46984	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
12	10	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑋 ⊆ ℂ)
13	6	faccld 14316	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
14	13	nncnd 12244	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
15	14	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ)
16		fzfid 14005	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
17		fzssnn0 46055	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
18		ssrab2 4034	. . . . . . . . . . . . 13 ⊢ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁} ⊆ ((0...𝑁) ↑_m (0...𝑀))
19		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
20	8, 6	etransclem12 46980	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁})
21	20	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁})
22	19, 21	eleqtrd 2865	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁})
23	18, 22	sselid 3935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
24		elmapi 8842	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
25	23, 24	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐:(0...𝑀)⟶(0...𝑁))
26	25	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ (0...𝑁))
27	17, 26	sselid 3935	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℕ₀)
28	27	faccld 14316	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℕ)
29	28	nncnd 12244	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℂ)
30	16, 29	fprodcl 16002	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ∈ ℂ)
31	28	nnne0d 12281	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ≠ 0)
32	16, 29, 31	fprodn0 16029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ≠ 0)
33	15, 30, 32	divcld 11986	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) ∈ ℂ)
34		ssid 3959	. . . . . 6 ⊢ ℂ ⊆ ℂ
35	34	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ℂ ⊆ ℂ)
36	12, 33, 35	constcncfg 46606	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))) ∈ (𝑋–cn→ℂ))
37	1	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
38	2	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
39	3	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
40		etransclem5 46973	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
41	7, 40	eqtri 2786	. . . . . . . 8 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
42		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ (0...𝑀))
43	37, 38, 39, 41, 42, 27	etransclem20 46988	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ)
44	43	3adant2 1149	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ)
45		simp2 1155	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → 𝑥 ∈ 𝑋)
46	44, 45	ffvelcdmd 7080	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥) ∈ ℂ)
47	43	feqmptd 6949	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)))
48	37, 38, 39, 41, 42, 27	etransclem22 46990	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)) ∈ (𝑋–cn→ℂ))
49	47, 48	eqeltrrd 2864	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑥 ∈ 𝑋 ↦ (((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)) ∈ (𝑋–cn→ℂ))
50	12, 16, 46, 49	fprodcncf 46634	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)) ∈ (𝑋–cn→ℂ))
51	36, 50	mulcncf 25605	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ (((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) · ∏𝑘 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))) ∈ (𝑋–cn→ℂ))
52	10, 11, 51	fsumcncf 46612	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) · ∏𝑘 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))) ∈ (𝑋–cn→ℂ))
53	9, 52	eqeltrd 2863	1 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 {crab 3416 ⊆ wss 3905 ifcif 4487 {cpr 4591 ↦ cmpt 5192 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8820 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 · cmul 11100 − cmin 11436 / cdiv 11866 ℕcn 12228 ℕ₀cn0 12499 ...cfz 13530 ↑cexp 14093 !cfa 14305 Σcsu 15733 ∏cprod 15953 ↾_t crest 17468 TopOpenctopn 17469 ℂ_fldccnfld 21522 –cn→ccncf 25035 D^𝑛 cdvn 26023

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-prod 15954 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026 df-dvn 26027

This theorem is referenced by: etransclem40 47008

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