etransclem29 - Mathbox for Glauco Siliprandi

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Theorem etransclem29 46268

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 45941	. . . . 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 46243	. . . 4 ⊢ (𝜑 → 𝐹 = 𝐸)
10	9	oveq2d 7406	. . 3 ⊢ (𝜑 → (𝑆 D^𝑛 𝐹) = (𝑆 D^𝑛 𝐸))
11	10	fveq1d 6863	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = ((𝑆 D^𝑛 𝐸)‘𝑁))
12		fzfid 13945	. . 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 46240	. . 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 46244	. . . . 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 2753	. . . 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 13588	. . . . 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 46259	. . 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 45954	. 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 2765	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 3408 ⊆ wss 3917 ifcif 4491 {cpr 4594 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 − cmin 11412 / cdiv 11842 ℕcn 12193 ℕ₀cn0 12449 ...cfz 13475 ↑cexp 14033 !cfa 14245 Σcsu 15659 ∏cprod 15876 ↾_t crest 17390 TopOpenctopn 17391 ℂ_fldccnfld 21271 D^𝑛 cdvn 25772

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-prod 15877 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-dvn 25776

This theorem is referenced by: etransclem30 46269

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