etransclem45 - Mathbox for Glauco Siliprandi

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Theorem etransclem45 43495

Description: 𝐾 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem45.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem45.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem45.f	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem45.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
etransclem45.k	⊢ 𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1)))

**Assertion**
Ref	Expression
etransclem45	⊢ (𝜑 → 𝐾 ∈ ℤ)

Distinct variable groups: 𝑗,𝑀,𝑘,𝑥 𝑃,𝑗,𝑘,𝑥 𝑅,𝑗,𝑘,𝑥 𝜑,𝑗,𝑘,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑗,𝑘) 𝐹(𝑥,𝑗,𝑘) 𝐾(𝑥,𝑗,𝑘)

Proof of Theorem etransclem45 Dummy variables 𝑐 𝑑 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem45.k	. 2 ⊢ 𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1)))
2		fzfi 13545	. . . . . 6 ⊢ (0...𝑀) ∈ Fin
3		fzfi 13545	. . . . . 6 ⊢ (0...𝑅) ∈ Fin
4		xpfi 8942	. . . . . 6 ⊢ (((0...𝑀) ∈ Fin ∧ (0...𝑅) ∈ Fin) → ((0...𝑀) × (0...𝑅)) ∈ Fin)
5	2, 3, 4	mp2an 692	. . . . 5 ⊢ ((0...𝑀) × (0...𝑅)) ∈ Fin
6	5	a1i 11	. . . 4 ⊢ (𝜑 → ((0...𝑀) × (0...𝑅)) ∈ Fin)
7		etransclem45.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
8		nnm1nn0 12131	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
10	9	faccld 13850	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
11	10	nncnd 11846	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
12		etransclem45.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
13	12	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝐴:ℕ₀⟶ℤ)
14		xp1st 7793	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (1^st ‘𝑘) ∈ (0...𝑀))
15		elfznn0 13205	. . . . . . . . 9 ⊢ ((1^st ‘𝑘) ∈ (0...𝑀) → (1^st ‘𝑘) ∈ ℕ₀)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (1^st ‘𝑘) ∈ ℕ₀)
17	16	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℕ₀)
18	13, 17	ffvelrnd 6905	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
19	18	zcnd 12283	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
20		reelprrecn 10821	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ℝ ∈ {ℝ, ℂ})
22		reopn 42500	. . . . . . . . 9 ⊢ ℝ ∈ (topGen‘ran (,))
23		eqid 2737	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24	23	tgioo2 23700	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
25	22, 24	eleqtri 2836	. . . . . . . 8 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
26	25	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
27	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝑃 ∈ ℕ)
28		etransclem45.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
29	28	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝑀 ∈ ℕ₀)
30		etransclem45.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
31		xp2nd 7794	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ (0...𝑅))
32		elfznn0 13205	. . . . . . . . 9 ⊢ ((2^nd ‘𝑘) ∈ (0...𝑅) → (2^nd ‘𝑘) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ ℕ₀)
34	33	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (2^nd ‘𝑘) ∈ ℕ₀)
35	21, 26, 27, 29, 30, 34	etransclem33 43483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
36	17	nn0red 12151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℝ)
37	35, 36	ffvelrnd 6905	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
38	19, 37	mulcld 10853	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℂ)
39	10	nnne0d 11880	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
40	6, 11, 38, 39	fsumdivc 15350	. . 3 ⊢ (𝜑 → (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) = Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))(((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
41	11	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℂ)
42	39	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ≠ 0)
43	19, 37, 41, 42	divassd 11643	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) = ((𝐴‘(1^st ‘𝑘)) · ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1)))))
44		etransclem5 43455	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ ℝ ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ ℝ ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
45		etransclem11 43461	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
46	14	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ (0...𝑀))
47	21, 26, 27, 29, 30, 34, 44, 45, 46, 36	etransclem37 43487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))
48	10	nnzd 12281	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
49	48	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℤ)
50	17	nn0zd 12280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℤ)
51	21, 26, 27, 29, 30, 34, 36, 50	etransclem42 43492	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
52		dvdsval2 15818	. . . . . . . 8 ⊢ (((!‘(𝑃 − 1)) ∈ ℤ ∧ (!‘(𝑃 − 1)) ≠ 0 ∧ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ) → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ↔ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ))
53	49, 42, 51, 52	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ↔ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ))
54	47, 53	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ)
55	18, 54	zmulcld 12288	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((𝐴‘(1^st ‘𝑘)) · ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1)))) ∈ ℤ)
56	43, 55	eqeltrd 2838	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
57	6, 56	fsumzcl 15299	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))(((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
58	40, 57	eqeltrd 2838	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
59	1, 58	eqeltrid 2842	1 ⊢ (𝜑 → 𝐾 ∈ ℤ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 {crab 3065 ifcif 4439 {cpr 4543 class class class wbr 5053 ↦ cmpt 5135 × cxp 5549 ran crn 5552 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 1^st c1st 7759 2^nd c2nd 7760 ↑_m cmap 8508 Fincfn 8626 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 · cmul 10734 − cmin 11062 / cdiv 11489 ℕcn 11830 ℕ₀cn0 12090 ℤcz 12176 (,)cioo 12935 ...cfz 13095 ↑cexp 13635 !cfa 13839 Σcsu 15249 ∏cprod 15467 ∥ cdvds 15815 ↾_t crest 16925 TopOpenctopn 16926 topGenctg 16942 ℂ_fldccnfld 20363 D^𝑛 cdvn 24761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 df-prod 15468 df-dvds 15816 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 df-dvn 24765

This theorem is referenced by: etransclem47 43497

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