etransclem19 - Mathbox for Glauco Siliprandi

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Theorem etransclem19 43283

Description: The 𝑁-th derivative of 𝐻 is 0 if 𝑁 is large enough. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem19.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem19.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_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)

**Assertion**
Ref	Expression
etransclem19	⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Distinct variable groups: 𝑗,𝐽,𝑥 𝑗,𝑀,𝑥 𝑥,𝑁 𝑃,𝑗,𝑥 𝑥,𝑆 𝑗,𝑋,𝑥 𝜑,𝑗,𝑥 Allowed substitution hints: 𝑆(𝑗) 𝐻(𝑥,𝑗) 𝑁(𝑗)

**Proof of Theorem etransclem19**
Step	Hyp	Ref	Expression
1		etransclem19.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		etransclem19.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_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 10682	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
8	6	zred 12126	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		nnm1nn0 11975	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
11	10	nn0red 11995	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
12	3	nnred 11689	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13	11, 12	ifcld 4466	. . . . . 6 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
14	10	nn0ge0d 11997	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
15	14	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ (𝑃 − 1))
16		iftrue 4426	. . . . . . . . . 10 ⊢ (𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
17	16	eqcomd 2764	. . . . . . . . 9 ⊢ (𝐽 = 0 → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
18	17	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
19	15, 18	breqtrd 5058	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
20	3	nnnn0d 11994	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
21	20	nn0ge0d 11997	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑃)
22	21	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ 𝑃)
23		iffalse 4429	. . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = 𝑃)
24	23	eqcomd 2764	. . . . . . . . 9 ⊢ (¬ 𝐽 = 0 → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
25	24	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
26	22, 25	breqtrd 5058	. . . . . . 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 10836	. . . . 5 ⊢ (𝜑 → 0 < 𝑁)
30	7, 8, 29	ltled 10826	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
31		elnn0z 12033	. . . 4 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
32	6, 30, 31	sylanbrc 586	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
33	1, 2, 3, 4, 5, 32	etransclem17 43281	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 (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 4428	. . 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 5128	. 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 2793	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ifcif 4420 {cpr 4524 class class class wbr 5032 ↦ cmpt 5112 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 ℝcr 10574 0cc0 10575 1c1 10576 · cmul 10580 < clt 10713 ≤ cle 10714 − cmin 10908 / cdiv 11335 ℕcn 11674 ℕ₀cn0 11934 ℤcz 12020 ...cfz 12939 ↑cexp 13479 !cfa 13683 ↾_t crest 16752 TopOpenctopn 16753 ℂ_fldccnfld 20166 D^𝑛 cdvn 24563

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653 ax-addf 10654 ax-mulf 10655

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7405 df-om 7580 df-1st 7693 df-2nd 7694 df-supp 7836 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-2o 8113 df-er 8299 df-map 8418 df-pm 8419 df-ixp 8480 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fsupp 8867 df-fi 8908 df-sup 8939 df-inf 8940 df-oi 9007 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-icc 12786 df-fz 12940 df-fzo 13083 df-seq 13419 df-exp 13480 df-fac 13684 df-hash 13741 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-starv 16638 df-sca 16639 df-vsca 16640 df-ip 16641 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-hom 16647 df-cco 16648 df-rest 16754 df-topn 16755 df-0g 16773 df-gsum 16774 df-topgen 16775 df-pt 16776 df-prds 16779 df-xrs 16833 df-qtop 16838 df-imas 16839 df-xps 16841 df-mre 16915 df-mrc 16916 df-acs 16918 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-submnd 18023 df-mulg 18292 df-cntz 18514 df-cmn 18975 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-fbas 20163 df-fg 20164 df-cnfld 20167 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-cld 21719 df-ntr 21720 df-cls 21721 df-nei 21798 df-lp 21836 df-perf 21837 df-cn 21927 df-cnp 21928 df-haus 22015 df-tx 22262 df-hmeo 22455 df-fil 22546 df-fm 22638 df-flim 22639 df-flf 22640 df-xms 23022 df-ms 23023 df-tms 23024 df-cncf 23579 df-limc 24565 df-dv 24566 df-dvn 24567

This theorem is referenced by: etransclem32 43296

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