etransclem19 - Mathbox for Glauco Siliprandi

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Theorem etransclem19 40788

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 10079	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
8	6	zred 11520	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		nnm1nn0 11372	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
11	10	nn0red 11390	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
12	3	nnred 11073	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13	11, 12	ifcld 4164	. . . . . 6 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
14	10	nn0ge0d 11392	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ (𝑃 − 1))
16		iftrue 4125	. . . . . . . . . 10 ⊢ (𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
17	16	eqcomd 2657	. . . . . . . . 9 ⊢ (𝐽 = 0 → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
19	15, 18	breqtrd 4711	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
20	3	nnnn0d 11389	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
21	20	nn0ge0d 11392	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑃)
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ 𝑃)
23		iffalse 4128	. . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = 𝑃)
24	23	eqcomd 2657	. . . . . . . . 9 ⊢ (¬ 𝐽 = 0 → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
25	24	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
26	22, 25	breqtrd 4711	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
27	19, 26	pm2.61dan 849	. . . . . 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 10233	. . . . 5 ⊢ (𝜑 → 0 < 𝑁)
30	7, 8, 29	ltled 10223	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
31		elnn0z 11428	. . . 4 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
32	6, 30, 31	sylanbrc 699	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
33	1, 2, 3, 4, 5, 32	etransclem17 40786	. 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 4127	. . 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 4778	. 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 2685	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ifcif 4119 {cpr 4212 class class class wbr 4685 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 · cmul 9979 < clt 10112 ≤ cle 10113 − cmin 10304 / cdiv 10722 ℕcn 11058 ℕ₀cn0 11330 ℤcz 11415 ...cfz 12364 ↑cexp 12900 !cfa 13100 ↾_t crest 16128 TopOpenctopn 16129 ℂ_fldccnfld 19794 D^𝑛 cdvn 23673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-icc 12220 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-fac 13101 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-dvn 23677

This theorem is referenced by: etransclem32 40801

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