etransclem34 - Mathbox for Glauco Siliprandi

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Theorem etransclem34 43699

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 43695	. 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 43367	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
11	8, 6	etransclem16 43681	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
12	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑋 ⊆ ℂ)
13	6	faccld 13926	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
14	13	nncnd 11919	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ)
16		fzfid 13621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
17		fzssnn0 42746	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
18		ssrab2 4009	. . . . . . . . . . . . 13 ⊢ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁} ⊆ ((0...𝑁) ↑_m (0...𝑀))
19		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
20	8, 6	etransclem12 43677	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁})
21	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁})
22	19, 21	eleqtrd 2841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁})
23	18, 22	sselid 3915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
24		elmapi 8595	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐:(0...𝑀)⟶(0...𝑁))
26	25	ffvelrnda 6943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ (0...𝑁))
27	17, 26	sselid 3915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℕ₀)
28	27	faccld 13926	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℕ)
29	28	nncnd 11919	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℂ)
30	16, 29	fprodcl 15590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ∈ ℂ)
31	28	nnne0d 11953	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ≠ 0)
32	16, 29, 31	fprodn0 15617	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ≠ 0)
33	15, 30, 32	divcld 11681	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) ∈ ℂ)
34		ssid 3939	. . . . . 6 ⊢ ℂ ⊆ ℂ
35	34	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ℂ ⊆ ℂ)
36	12, 33, 35	constcncfg 43303	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))) ∈ (𝑋–cn→ℂ))
37	1	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
38	2	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
39	3	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
40		etransclem5 43670	. . . . . . . . 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 2766	. . . . . . . 8 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
42		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ (0...𝑀))
43	37, 38, 39, 41, 42, 27	etransclem20 43685	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ)
44	43	3adant2 1129	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ)
45		simp2 1135	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → 𝑥 ∈ 𝑋)
46	44, 45	ffvelrnd 6944	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥) ∈ ℂ)
47	43	feqmptd 6819	. . . . . 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 43687	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)) ∈ (𝑋–cn→ℂ))
49	47, 48	eqeltrrd 2840	. . . . 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 43331	. . . 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 24515	. . 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 43309	. 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 2839	1 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 {crab 3067 ⊆ wss 3883 ifcif 4456 {cpr 4560 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 − cmin 11135 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ...cfz 13168 ↑cexp 13710 !cfa 13915 Σcsu 15325 ∏cprod 15543 ↾_t crest 17048 TopOpenctopn 17049 ℂ_fldccnfld 20510 –cn→ccncf 23945 D^𝑛 cdvn 24933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-prod 15544 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-dvn 24937

This theorem is referenced by: etransclem40 43705

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