etransclem19 - Mathbox for Glauco Siliprandi

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Theorem etransclem19 43794

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 10978	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
8	6	zred 12426	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		nnm1nn0 12274	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
11	10	nn0red 12294	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
12	3	nnred 11988	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13	11, 12	ifcld 4505	. . . . . 6 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
14	10	nn0ge0d 12296	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
15	14	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ (𝑃 − 1))
16		iftrue 4465	. . . . . . . . . 10 ⊢ (𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
17	16	eqcomd 2744	. . . . . . . . 9 ⊢ (𝐽 = 0 → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
18	17	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
19	15, 18	breqtrd 5100	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
20	3	nnnn0d 12293	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
21	20	nn0ge0d 12296	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑃)
22	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ 𝑃)
23		iffalse 4468	. . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = 𝑃)
24	23	eqcomd 2744	. . . . . . . . 9 ⊢ (¬ 𝐽 = 0 → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
25	24	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
26	22, 25	breqtrd 5100	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
27	19, 26	pm2.61dan 810	. . . . . 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 11133	. . . . 5 ⊢ (𝜑 → 0 < 𝑁)
30	7, 8, 29	ltled 11123	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
31		elnn0z 12332	. . . 4 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
32	6, 30, 31	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
33	1, 2, 3, 4, 5, 32	etransclem17 43792	. 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 4467	. . 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 5176	. 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 2778	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ifcif 4459 {cpr 4563 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ...cfz 13239 ↑cexp 13782 !cfa 13987 ↾_t crest 17131 TopOpenctopn 17132 ℂ_fldccnfld 20597 D^𝑛 cdvn 25028

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-fac 13988 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-dvn 25032

This theorem is referenced by: etransclem32 43807

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