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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem34 | Structured version Visualization version GIF version |
Description: The 𝑁-th derivative of 𝐹 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem34.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
etransclem34.a | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾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}) |
Ref | Expression |
---|---|
etransclem34 | ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem34.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | etransclem34.a | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾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 46220 | . 2 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) · ∏𝑘 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)))) |
10 | 1, 2 | dvdmsscn 45892 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
11 | 8, 6 | etransclem16 46206 | . . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin) |
12 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑋 ⊆ ℂ) |
13 | 6 | faccld 14320 | . . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ) |
14 | 13 | nncnd 12280 | . . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ) |
15 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ) |
16 | fzfid 14011 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin) | |
17 | fzssnn0 45268 | . . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ0 | |
18 | ssrab2 4090 | . . . . . . . . . . . . 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 46202 | . . . . . . . . . . . . . . 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 3993 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀))) |
24 | elmapi 8888 | . . . . . . . . . . . 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 | ffvelcdmda 7104 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ (0...𝑁)) |
27 | 17, 26 | sselid 3993 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℕ0) |
28 | 27 | faccld 14320 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℕ) |
29 | 28 | nncnd 12280 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℂ) |
30 | 16, 29 | fprodcl 15985 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ∈ ℂ) |
31 | 28 | nnne0d 12314 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ≠ 0) |
32 | 16, 29, 31 | fprodn0 16012 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ≠ 0) |
33 | 15, 30, 32 | divcld 12041 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) ∈ ℂ) |
34 | ssid 4018 | . . . . . 6 ⊢ ℂ ⊆ ℂ | |
35 | 34 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ℂ ⊆ ℂ) |
36 | 12, 33, 35 | constcncfg 45828 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))) ∈ (𝑋–cn→ℂ)) |
37 | 1 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ}) |
38 | 2 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
39 | 3 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑃 ∈ ℕ) |
40 | etransclem5 46195 | . . . . . . . . 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 2763 | . . . . . . . 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 46210 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ) |
44 | 43 | 3adant2 1130 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ) |
45 | simp2 1136 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → 𝑥 ∈ 𝑋) | |
46 | 44, 45 | ffvelcdmd 7105 | . . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥) ∈ ℂ) |
47 | 43 | feqmptd 6977 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))) |
48 | 37, 38, 39, 41, 42, 27 | etransclem22 46212 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)) ∈ (𝑋–cn→ℂ)) |
49 | 47, 48 | eqeltrrd 2840 | . . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑥 ∈ 𝑋 ↦ (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)) ∈ (𝑋–cn→ℂ)) |
50 | 12, 16, 46, 49 | fprodcncf 45856 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)) ∈ (𝑋–cn→ℂ)) |
51 | 36, 50 | mulcncf 25494 | . . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ (((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) · ∏𝑘 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))) ∈ (𝑋–cn→ℂ)) |
52 | 10, 11, 51 | fsumcncf 45834 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) · ∏𝑘 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))) ∈ (𝑋–cn→ℂ)) |
53 | 9, 52 | eqeltrd 2839 | 1 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 ifcif 4531 {cpr 4633 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 − cmin 11490 / cdiv 11918 ℕcn 12264 ℕ0cn0 12524 ...cfz 13544 ↑cexp 14099 !cfa 14309 Σcsu 15719 ∏cprod 15936 ↾t crest 17467 TopOpenctopn 17468 ℂfldccnfld 21382 –cn→ccncf 24916 D𝑛 cdvn 25914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-prod 15937 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-dvn 25918 |
This theorem is referenced by: etransclem40 46230 |
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