etransclem29 - Mathbox for Glauco Siliprandi

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Theorem etransclem29 44148

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 43821	. . . . 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 44123	. . . 4 ⊢ (𝜑 → 𝐹 = 𝐸)
10	9	oveq2d 7353	. . 3 ⊢ (𝜑 → (𝑆 D^𝑛 𝐹) = (𝑆 D^𝑛 𝐸))
11	10	fveq1d 6827	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = ((𝑆 D^𝑛 𝐸)‘𝑁))
12		fzfid 13794	. . 3 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
13	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ⊆ ℂ)
14	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
15		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
16	13, 14, 7, 15	etransclem1 44120	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐻‘𝑗):𝑋⟶ℂ)
17		etransclem29.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18	1	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
19	2	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
20	4	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑃 ∈ ℕ)
21		etransclem5 44124	. . . . 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 2764	. . . 4 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
23		simp2 1136	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑀))
24		elfznn0 13450	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) → 𝑖 ∈ ℕ₀)
25	24	3ad2ant3 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑖 ∈ ℕ₀)
26	18, 19, 20, 22, 23, 25	etransclem20 44139	. . 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 43834	. 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 2776	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 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 {crab 3403 ⊆ wss 3898 ifcif 4473 {cpr 4575 ↦ cmpt 5175 ‘cfv 6479 (class class class)co 7337 ↑_m cmap 8686 ℂcc 10970 ℝcr 10971 0cc0 10972 1c1 10973 · cmul 10977 − cmin 11306 / cdiv 11733 ℕcn 12074 ℕ₀cn0 12334 ...cfz 13340 ↑cexp 13883 !cfa 14088 Σcsu 15496 ∏cprod 15714 ↾_t crest 17228 TopOpenctopn 17229 ℂ_fldccnfld 20703 D^𝑛 cdvn 25134

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-inf2 9498 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 ax-addf 11051 ax-mulf 11052

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-2o 8368 df-er 8569 df-map 8688 df-pm 8689 df-ixp 8757 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-fi 9268 df-sup 9299 df-inf 9300 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-q 12790 df-rp 12832 df-xneg 12949 df-xadd 12950 df-xmul 12951 df-ico 13186 df-icc 13187 df-fz 13341 df-fzo 13484 df-seq 13823 df-exp 13884 df-fac 14089 df-bc 14118 df-hash 14146 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-clim 15296 df-sum 15497 df-prod 15715 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-starv 17074 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-unif 17082 df-hom 17083 df-cco 17084 df-rest 17230 df-topn 17231 df-0g 17249 df-gsum 17250 df-topgen 17251 df-pt 17252 df-prds 17255 df-xrs 17310 df-qtop 17315 df-imas 17316 df-xps 17318 df-mre 17392 df-mrc 17393 df-acs 17395 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-mulg 18797 df-cntz 19019 df-cmn 19483 df-psmet 20695 df-xmet 20696 df-met 20697 df-bl 20698 df-mopn 20699 df-fbas 20700 df-fg 20701 df-cnfld 20704 df-top 22149 df-topon 22166 df-topsp 22188 df-bases 22202 df-cld 22276 df-ntr 22277 df-cls 22278 df-nei 22355 df-lp 22393 df-perf 22394 df-cn 22484 df-cnp 22485 df-haus 22572 df-tx 22819 df-hmeo 23012 df-fil 23103 df-fm 23195 df-flim 23196 df-flf 23197 df-xms 23579 df-ms 23580 df-tms 23581 df-cncf 24147 df-limc 25136 df-dv 25137 df-dvn 25138

This theorem is referenced by: etransclem30 44149

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