etransclem29 - Mathbox for Glauco Siliprandi

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Theorem etransclem29 46371

Description: The 𝑁-th derivative of 𝐹. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etranslemdvnf2lemlem.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem29.a	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem29.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem29.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem29.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem29.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
etransclem29.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem29.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
etransclem29.e	⊢ 𝐸 = (𝑥 ∈ 𝑋 ↦ ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥))

**Assertion**
Ref	Expression
etransclem29	⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥))))

Distinct variable groups: 𝐶,𝑐 𝐻,𝑐,𝑗,𝑛,𝑥 𝑀,𝑐,𝑗,𝑛,𝑥 𝑁,𝑐,𝑗,𝑛,𝑥 𝑃,𝑗,𝑥 𝑆,𝑐,𝑗,𝑛,𝑥 𝑗,𝑋,𝑥,𝑛 𝜑,𝑗,𝑥,𝑛 Allowed substitution hints: 𝜑(𝑐) 𝐶(𝑥,𝑗,𝑛) 𝑃(𝑛,𝑐) 𝐸(𝑥,𝑗,𝑛,𝑐) 𝐹(𝑥,𝑗,𝑛,𝑐) 𝑋(𝑐)

Proof of Theorem etransclem29 Dummy variables 𝑖 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etranslemdvnf2lemlem.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		etransclem29.a	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
3	1, 2	dvdmsscn 46044	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
4		etransclem29.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		etransclem29.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
6		etransclem29.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
7		etransclem29.h	. . . . 5 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
8		etransclem29.e	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ 𝑋 ↦ ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥))
9	3, 4, 5, 6, 7, 8	etransclem4 46346	. . . 4 ⊢ (𝜑 → 𝐹 = 𝐸)
10	9	oveq2d 7362	. . 3 ⊢ (𝜑 → (𝑆 D^𝑛 𝐹) = (𝑆 D^𝑛 𝐸))
11	10	fveq1d 6824	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = ((𝑆 D^𝑛 𝐸)‘𝑁))
12		fzfid 13880	. . 3 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
13	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ⊆ ℂ)
14	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
15		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
16	13, 14, 7, 15	etransclem1 46343	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐻‘𝑗):𝑋⟶ℂ)
17		etransclem29.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18	1	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
19	2	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
20	4	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑃 ∈ ℕ)
21		etransclem5 46347	. . . . 5 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
22	7, 21	eqtri 2754	. . . 4 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
23		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑀))
24		elfznn0 13520	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) → 𝑖 ∈ ℕ₀)
25	24	3ad2ant3 1135	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑖 ∈ ℕ₀)
26	18, 19, 20, 22, 23, 25	etransclem20 46362	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑗))‘𝑖):𝑋⟶ℂ)
27		etransclem29.c	. . 3 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
28	1, 2, 12, 16, 17, 26, 8, 27	dvnprod 46057	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐸)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥))))
29	11, 28	eqtrd 2766	1 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 {crab 3395 ⊆ wss 3897 ifcif 4472 {cpr 4575 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 · cmul 11011 − cmin 11344 / cdiv 11774 ℕcn 12125 ℕ₀cn0 12381 ...cfz 13407 ↑cexp 13968 !cfa 14180 Σcsu 15593 ∏cprod 15810 ↾_t crest 17324 TopOpenctopn 17325 ℂ_fldccnfld 21291 D^𝑛 cdvn 25792

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-prod 15811 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-dvn 25796

This theorem is referenced by: etransclem30 46372

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