etransclem45 - Mathbox for Glauco Siliprandi

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Theorem etransclem45 45080

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 13939	. . . . . 6 ⊢ (0...𝑀) ∈ Fin
3		fzfi 13939	. . . . . 6 ⊢ (0...𝑅) ∈ Fin
4		xpfi 9319	. . . . . 6 ⊢ (((0...𝑀) ∈ Fin ∧ (0...𝑅) ∈ Fin) → ((0...𝑀) × (0...𝑅)) ∈ Fin)
5	2, 3, 4	mp2an 690	. . . . 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 12515	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
10	9	faccld 14246	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
11	10	nncnd 12230	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
12		etransclem45.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝐴:ℕ₀⟶ℤ)
14		xp1st 8009	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (1^st ‘𝑘) ∈ (0...𝑀))
15		elfznn0 13596	. . . . . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℕ₀)
18	13, 17	ffvelcdmd 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
19	18	zcnd 12669	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
20		reelprrecn 11204	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ℝ ∈ {ℝ, ℂ})
22		reopn 44084	. . . . . . . . 9 ⊢ ℝ ∈ (topGen‘ran (,))
23		eqid 2732	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24	23	tgioo2 24326	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
25	22, 24	eleqtri 2831	. . . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝑃 ∈ ℕ)
28		etransclem45.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
29	28	adantr 481	. . . . . . 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 8010	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ (0...𝑅))
32		elfznn0 13596	. . . . . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (2^nd ‘𝑘) ∈ ℕ₀)
35	21, 26, 27, 29, 30, 34	etransclem33 45068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
36	17	nn0red 12535	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℝ)
37	35, 36	ffvelcdmd 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
38	19, 37	mulcld 11236	. . . 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 12264	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
40	6, 11, 38, 39	fsumdivc 15734	. . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℂ)
42	39	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ≠ 0)
43	19, 37, 41, 42	divassd 12027	. . . . 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 45040	. . . . . . . 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 45046	. . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ (0...𝑀))
47	21, 26, 27, 29, 30, 34, 44, 45, 46, 36	etransclem37 45072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))
48	10	nnzd 12587	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
49	48	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℤ)
50	17	nn0zd 12586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℤ)
51	21, 26, 27, 29, 30, 34, 36, 50	etransclem42 45077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
52		dvdsval2 16202	. . . . . . . 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 1371	. . . . . . 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 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ)
55	18, 54	zmulcld 12674	. . . . 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 2833	. . . 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 15683	. . 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 2833	. 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 2837	1 ⊢ (𝜑 → 𝐾 ∈ ℤ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 {crab 3432 ifcif 4528 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 ↑_m cmap 8822 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 − cmin 11446 / cdiv 11873 ℕcn 12214 ℕ₀cn0 12474 ℤcz 12560 (,)cioo 13326 ...cfz 13486 ↑cexp 14029 !cfa 14235 Σcsu 15634 ∏cprod 15851 ∥ cdvds 16199 ↾_t crest 17368 TopOpenctopn 17369 topGenctg 17385 ℂ_fldccnfld 20950 D^𝑛 cdvn 25388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-prod 15852 df-dvds 16200 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-limc 25390 df-dv 25391 df-dvn 25392

This theorem is referenced by: etransclem47 45082

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