etransclem29 - Mathbox for Glauco Siliprandi

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Theorem etransclem29 46259

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 45932	. . . . 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 46234	. . . 4 ⊢ (𝜑 → 𝐹 = 𝐸)
10	9	oveq2d 7426	. . 3 ⊢ (𝜑 → (𝑆 D^𝑛 𝐹) = (𝑆 D^𝑛 𝐸))
11	10	fveq1d 6883	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = ((𝑆 D^𝑛 𝐸)‘𝑁))
12		fzfid 13996	. . 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 46231	. . 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 46235	. . . . 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 2759	. . . 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 13642	. . . . 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 46250	. . 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 45945	. 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 2771	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 1540 ∈ wcel 2109 {crab 3420 ⊆ wss 3931 ifcif 4505 {cpr 4608 ↦ cmpt 5206 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 · cmul 11139 − cmin 11471 / cdiv 11899 ℕcn 12245 ℕ₀cn0 12506 ...cfz 13529 ↑cexp 14084 !cfa 14296 Σcsu 15707 ∏cprod 15924 ↾_t crest 17439 TopOpenctopn 17440 ℂ_fldccnfld 21320 D^𝑛 cdvn 25822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-sum 15708 df-prod 15925 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825 df-dvn 25826

This theorem is referenced by: etransclem30 46260

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