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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem19 | Structured version Visualization version GIF version |
Description: The 𝑁-th derivative of 𝐻 is 0 if 𝑁 is large enough. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem19.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
etransclem19.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
etransclem19.p | ⊢ (𝜑 → 𝑃 ∈ ℕ) |
etransclem19.1 | ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) |
etransclem19.J | ⊢ (𝜑 → 𝐽 ∈ (0...𝑀)) |
etransclem19.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
etransclem19.7 | ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) |
Ref | Expression |
---|---|
etransclem19 | ⊢ (𝜑 → ((𝑆 D𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem19.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | etransclem19.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
3 | etransclem19.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ) | |
4 | etransclem19.1 | . . 3 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) | |
5 | etransclem19.J | . . 3 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀)) | |
6 | etransclem19.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
7 | 0red 11204 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ) | |
8 | 6 | zred 12653 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
9 | nnm1nn0 12500 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0) |
11 | 10 | nn0red 12520 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ) |
12 | 3 | nnred 12214 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
13 | 11, 12 | ifcld 4570 | . . . . . 6 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ) |
14 | 10 | nn0ge0d 12522 | . . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑃 − 1)) |
15 | 14 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ (𝑃 − 1)) |
16 | iftrue 4530 | . . . . . . . . . 10 ⊢ (𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1)) | |
17 | 16 | eqcomd 2739 | . . . . . . . . 9 ⊢ (𝐽 = 0 → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
18 | 17 | adantl 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
19 | 15, 18 | breqtrd 5170 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
20 | 3 | nnnn0d 12519 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ0) |
21 | 20 | nn0ge0d 12522 | . . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑃) |
22 | 21 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ 𝑃) |
23 | iffalse 4533 | . . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = 𝑃) | |
24 | 23 | eqcomd 2739 | . . . . . . . . 9 ⊢ (¬ 𝐽 = 0 → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
25 | 24 | adantl 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
26 | 22, 25 | breqtrd 5170 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
27 | 19, 26 | pm2.61dan 812 | . . . . . 6 ⊢ (𝜑 → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
28 | etransclem19.7 | . . . . . 6 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) | |
29 | 7, 13, 8, 27, 28 | lelttrd 11359 | . . . . 5 ⊢ (𝜑 → 0 < 𝑁) |
30 | 7, 8, 29 | ltled 11349 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝑁) |
31 | elnn0z 12558 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) | |
32 | 6, 30, 31 | sylanbrc 584 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
33 | 1, 2, 3, 4, 5, 32 | etransclem17 44840 | . 2 ⊢ (𝜑 → ((𝑆 D𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))))) |
34 | 28 | iftrued 4532 | . . 3 ⊢ (𝜑 → if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) = 0) |
35 | 34 | mpteq2dv 5246 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) = (𝑥 ∈ 𝑋 ↦ 0)) |
36 | 33, 35 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝑆 D𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ifcif 4524 {cpr 4626 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 ℝcr 11096 0cc0 11097 1c1 11098 · cmul 11102 < clt 11235 ≤ cle 11236 − cmin 11431 / cdiv 11858 ℕcn 12199 ℕ0cn0 12459 ℤcz 12545 ...cfz 13471 ↑cexp 14014 !cfa 14220 ↾t crest 17353 TopOpenctopn 17354 ℂfldccnfld 20918 D𝑛 cdvn 25350 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 ax-addf 11176 ax-mulf 11177 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8691 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-fi 9393 df-sup 9424 df-inf 9425 df-oi 9492 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-icc 13318 df-fz 13472 df-fzo 13615 df-seq 13954 df-exp 14015 df-fac 14221 df-hash 14278 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-starv 17199 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-hom 17208 df-cco 17209 df-rest 17355 df-topn 17356 df-0g 17374 df-gsum 17375 df-topgen 17376 df-pt 17377 df-prds 17380 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-mulg 18936 df-cntz 19166 df-cmn 19634 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-fbas 20915 df-fg 20916 df-cnfld 20919 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-cld 22492 df-ntr 22493 df-cls 22494 df-nei 22571 df-lp 22609 df-perf 22610 df-cn 22700 df-cnp 22701 df-haus 22788 df-tx 23035 df-hmeo 23228 df-fil 23319 df-fm 23411 df-flim 23412 df-flf 23413 df-xms 23795 df-ms 23796 df-tms 23797 df-cncf 24363 df-limc 25352 df-dv 25353 df-dvn 25354 |
This theorem is referenced by: etransclem32 44855 |
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