etransclem19 - Mathbox for Glauco Siliprandi

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Theorem etransclem19 42558

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 10644	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
8	6	zred 12088	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		nnm1nn0 11939	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
11	10	nn0red 11957	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
12	3	nnred 11653	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13	11, 12	ifcld 4512	. . . . . 6 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
14	10	nn0ge0d 11959	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
15	14	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ (𝑃 − 1))
16		iftrue 4473	. . . . . . . . . 10 ⊢ (𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
17	16	eqcomd 2827	. . . . . . . . 9 ⊢ (𝐽 = 0 → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
18	17	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → (𝑃 − 1) = if(𝐽 = 0, (𝑃 − 1), 𝑃))
19	15, 18	breqtrd 5092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
20	3	nnnn0d 11956	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
21	20	nn0ge0d 11959	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑃)
22	21	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ 𝑃)
23		iffalse 4476	. . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → if(𝐽 = 0, (𝑃 − 1), 𝑃) = 𝑃)
24	23	eqcomd 2827	. . . . . . . . 9 ⊢ (¬ 𝐽 = 0 → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
25	24	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑃 = if(𝐽 = 0, (𝑃 − 1), 𝑃))
26	22, 25	breqtrd 5092	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 0 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
27	19, 26	pm2.61dan 811	. . . . . 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 10798	. . . . 5 ⊢ (𝜑 → 0 < 𝑁)
30	7, 8, 29	ltled 10788	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
31		elnn0z 11995	. . . 4 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
32	6, 30, 31	sylanbrc 585	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
33	1, 2, 3, 4, 5, 32	etransclem17 42556	. 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 4475	. . 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 5162	. 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 2856	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ifcif 4467 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 ℕcn 11638 ℕ₀cn0 11898 ℤcz 11982 ...cfz 12893 ↑cexp 13430 !cfa 13634 ↾_t crest 16694 TopOpenctopn 16695 ℂ_fldccnfld 20545 D^𝑛 cdvn 24462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-dvn 24466

This theorem is referenced by: etransclem32 42571

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