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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 44495 | . 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 44167 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
11 | 8, 6 | etransclem16 44481 | . . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin) |
12 | 10 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑋 ⊆ ℂ) |
13 | 6 | faccld 14184 | . . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ) |
14 | 13 | nncnd 12169 | . . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ) |
15 | 14 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ) |
16 | fzfid 13878 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin) | |
17 | fzssnn0 43541 | . . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ0 | |
18 | ssrab2 4037 | . . . . . . . . . . . . 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 44477 | . . . . . . . . . . . . . . 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 2840 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = 𝑁}) |
23 | 18, 22 | sselid 3942 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀))) |
24 | elmapi 8787 | . . . . . . . . . . . 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 7035 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ (0...𝑁)) |
27 | 17, 26 | sselid 3942 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℕ0) |
28 | 27 | faccld 14184 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℕ) |
29 | 28 | nncnd 12169 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ∈ ℂ) |
30 | 16, 29 | fprodcl 15835 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ∈ ℂ) |
31 | 28 | nnne0d 12203 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → (!‘(𝑐‘𝑘)) ≠ 0) |
32 | 16, 29, 31 | fprodn0 15862 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)) ≠ 0) |
33 | 15, 30, 32 | divcld 11931 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) ∈ ℂ) |
34 | ssid 3966 | . . . . . 6 ⊢ ℂ ⊆ ℂ | |
35 | 34 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ℂ ⊆ ℂ) |
36 | 12, 33, 35 | constcncfg 44103 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))) ∈ (𝑋–cn→ℂ)) |
37 | 1 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ}) |
38 | 2 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
39 | 3 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → 𝑃 ∈ ℕ) |
40 | etransclem5 44470 | . . . . . . . . 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 2764 | . . . . . . . 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 44485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ) |
44 | 43 | 3adant2 1131 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)):𝑋⟶ℂ) |
45 | simp2 1137 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → 𝑥 ∈ 𝑋) | |
46 | 44, 45 | ffvelcdmd 7036 | . . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥) ∈ ℂ) |
47 | 43 | feqmptd 6910 | . . . . . 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 44487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑘 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)) ∈ (𝑋–cn→ℂ)) |
49 | 47, 48 | eqeltrrd 2839 | . . . . 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 44131 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)) ∈ (𝑋–cn→ℂ)) |
51 | 36, 50 | mulcncf 24810 | . . 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 44109 | . 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 2838 | 1 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3407 ⊆ wss 3910 ifcif 4486 {cpr 4588 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 · cmul 11056 − cmin 11385 / cdiv 11812 ℕcn 12153 ℕ0cn0 12413 ...cfz 13424 ↑cexp 13967 !cfa 14173 Σcsu 15570 ∏cprod 15788 ↾t crest 17302 TopOpenctopn 17303 ℂfldccnfld 20796 –cn→ccncf 24239 D𝑛 cdvn 25228 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-prod 15789 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-dvn 25232 |
This theorem is referenced by: etransclem40 44505 |
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