etransclem45 - Mathbox for Glauco Siliprandi

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Theorem etransclem45 46284

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 13944	. . . . . 6 ⊢ (0...𝑀) ∈ Fin
3		fzfi 13944	. . . . . 6 ⊢ (0...𝑅) ∈ Fin
4		xpfi 9276	. . . . . 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 12490	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
10	9	faccld 14256	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
11	10	nncnd 12209	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
12		etransclem45.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝐴:ℕ₀⟶ℤ)
14		xp1st 8003	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (1^st ‘𝑘) ∈ (0...𝑀))
15		elfznn0 13588	. . . . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℕ₀)
18	13, 17	ffvelcdmd 7060	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
19	18	zcnd 12646	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
20		reelprrecn 11167	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ℝ ∈ {ℝ, ℂ})
22		reopn 45294	. . . . . . . . 9 ⊢ ℝ ∈ (topGen‘ran (,))
23		tgioo4 24700	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
24	22, 23	eleqtri 2827	. . . . . . . 8 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
26	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝑃 ∈ ℕ)
27		etransclem45.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
28	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝑀 ∈ ℕ₀)
29		etransclem45.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
30		xp2nd 8004	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ (0...𝑅))
31		elfznn0 13588	. . . . . . . . 9 ⊢ ((2^nd ‘𝑘) ∈ (0...𝑅) → (2^nd ‘𝑘) ∈ ℕ₀)
32	30, 31	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ ℕ₀)
33	32	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (2^nd ‘𝑘) ∈ ℕ₀)
34	21, 25, 26, 28, 29, 33	etransclem33 46272	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
35	17	nn0red 12511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℝ)
36	34, 35	ffvelcdmd 7060	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
37	19, 36	mulcld 11201	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℂ)
38	10	nnne0d 12243	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
39	6, 11, 37, 38	fsumdivc 15759	. . 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))))
40	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℂ)
41	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ≠ 0)
42	19, 36, 40, 41	divassd 12000	. . . . 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)))))
43		etransclem5 46244	. . . . . . . 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))))
44		etransclem11 46250	. . . . . . . 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})
45	14	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ (0...𝑀))
46	21, 25, 26, 28, 29, 33, 43, 44, 45, 35	etransclem37 46276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))
47	10	nnzd 12563	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
48	47	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℤ)
49	17	nn0zd 12562	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℤ)
50	21, 25, 26, 28, 29, 33, 35, 49	etransclem42 46281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
51		dvdsval2 16232	. . . . . . . 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))
52	48, 41, 50, 51	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))
53	46, 52	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ)
54	18, 53	zmulcld 12651	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((𝐴‘(1^st ‘𝑘)) · ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1)))) ∈ ℤ)
55	42, 54	eqeltrd 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
56	6, 55	fsumzcl 15708	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))(((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
57	39, 56	eqeltrd 2829	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
58	1, 57	eqeltrid 2833	1 ⊢ (𝜑 → 𝐾 ∈ ℤ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 {crab 3408 ifcif 4491 {cpr 4594 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ran crn 5642 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 ↑_m cmap 8802 Fincfn 8921 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 − cmin 11412 / cdiv 11842 ℕcn 12193 ℕ₀cn0 12449 ℤcz 12536 (,)cioo 13313 ...cfz 13475 ↑cexp 14033 !cfa 14245 Σcsu 15659 ∏cprod 15876 ∥ cdvds 16229 ↾_t crest 17390 TopOpenctopn 17391 topGenctg 17407 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-ioo 13317 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-dvds 16230 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: etransclem47 46286

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