etransclem29 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem29		Structured version Visualization version GIF version

Theorem etransclem29 47546

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 47219	. . . . 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 47521	. . . 4 ⊢ (𝜑 → 𝐹 = 𝐸)
10	9	oveq2d 7467	. . 3 ⊢ (𝜑 → (𝑆 D^𝑛 𝐹) = (𝑆 D^𝑛 𝐸))
11	10	fveq1d 6918	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = ((𝑆 D^𝑛 𝐸)‘𝑁))
12		fzfid 14156	. . 3 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
13	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ⊆ ℂ)
14	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
15		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
16	13, 14, 7, 15	etransclem1 47518	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐻‘𝑗):𝑋⟶ℂ)
17		etransclem29.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18	1	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
19	2	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
20	4	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑃 ∈ ℕ)
21		etransclem5 47522	. . . . 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 2786	. . . 4 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
23		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑀))
24		elfznn0 13794	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) → 𝑖 ∈ ℕ₀)
25	24	3ad2ant3 1153	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑁)) → 𝑖 ∈ ℕ₀)
26	18, 19, 20, 22, 23, 25	etransclem20 47537	. . 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 47232	. 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 2798	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {crab 3416 ⊆ wss 3900 ifcif 4485 {cpr 4590 ↦ cmpt 5192 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 · cmul 11244 − cmin 11580 / cdiv 12013 ℕcn 12375 ℕ₀cn0 12646 ...cfz 13679 ↑cexp 14244 !cfa 14458 Σcsu 15897 ∏cprod 16116 ↾_t crest 17641 TopOpenctopn 17642 ℂ_fldccnfld 21729 D^𝑛 cdvn 26239

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-sum 15898 df-prod 16117 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-dvn 26243

This theorem is used by: etransclem30 47547

W3C validator