etransclem34 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem34		Structured version Visualization version GIF version

Theorem etransclem34 43790

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 43786	. 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 43458	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
11	8, 6	etransclem16 43772	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
12	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑋 ⊆ ℂ)
13	6	faccld 14008	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
14	13	nncnd 11999	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ)
16		fzfid 13703	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
17		fzssnn0 42837	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
18		ssrab2 4012	. . . . . . . . . . . . 13 ⊢ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁} ⊆ ((0...𝑁) ↑_m (0...𝑀))
19		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
20	8, 6	etransclem12 43768	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁})
21	20	adantr 481	. . . . . . . . . . . . . 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 3918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
24		elmapi 8624	. . . . . . . . . . . 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 6953	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ (0...𝑁))
27	17, 26	sselid 3918	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℕ₀)
28	27	faccld 14008	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℕ)
29	28	nncnd 11999	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℂ)
30	16, 29	fprodcl 15672	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ∈ ℂ)
31	28	nnne0d 12033	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ≠ 0)
32	16, 29, 31	fprodn0 15699	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ≠ 0)
33	15, 30, 32	divcld 11761	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) ∈ ℂ)
34		ssid 3942	. . . . . 6 ⊢ ℂ ⊆ ℂ
35	34	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ℂ ⊆ ℂ)
36	12, 33, 35	constcncfg 43394	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))) ∈ (𝑋–cn→ℂ))
37	1	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
38	2	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
39	3	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
40		etransclem5 43761	. . . . . . . . 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 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ (0...𝑀))
43	37, 38, 39, 41, 42, 27	etransclem20 43776	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ)
44	43	3adant2 1130	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ)
45		simp2 1136	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → 𝑥 ∈ 𝑋)
46	44, 45	ffvelrnd 6954	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥) ∈ ℂ)
47	43	feqmptd 6829	. . . . . 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 43778	. . . . . 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 43422	. . . 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 24620	. . 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 43400	. 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3886 ifcif 4459 {cpr 4563 ↦ cmpt 5156 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 ↑_m cmap 8602 ℂcc 10879 ℝcr 10880 0cc0 10881 1c1 10882 · cmul 10886 − cmin 11215 / cdiv 11642 ℕcn 11983 ℕ₀cn0 12243 ...cfz 13249 ↑cexp 13792 !cfa 13997 Σcsu 15407 ∏cprod 15625 ↾_t crest 17141 TopOpenctopn 17142 ℂ_fldccnfld 20607 –cn→ccncf 24049 D^𝑛 cdvn 25038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-seq 13732 df-exp 13793 df-fac 13998 df-bc 14027 df-hash 14055 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-clim 15207 df-sum 15408 df-prod 15626 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-mulg 18711 df-cntz 18933 df-cmn 19398 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-lp 22297 df-perf 22298 df-cn 22388 df-cnp 22389 df-haus 22476 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-xms 23483 df-ms 23484 df-tms 23485 df-cncf 24051 df-limc 25040 df-dv 25041 df-dvn 25042

This theorem is referenced by: etransclem40 43796

W3C validator