etransclem45 - Mathbox for Glauco Siliprandi

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Theorem etransclem45 46885

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 14008	. . . . . 6 ⊢ (0...𝑀) ∈ Fin
3		fzfi 14008	. . . . . 6 ⊢ (0...𝑅) ∈ Fin
4		xpfi 9279	. . . . . 6 ⊢ (((0...𝑀) ∈ Fin ∧ (0...𝑅) ∈ Fin) → ((0...𝑀) × (0...𝑅)) ∈ Fin)
5	2, 3, 4	mp2an 704	. . . . 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 12545	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
9	7, 8	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
10	9	faccld 14320	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
11	10	nncnd 12249	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
12		etransclem45.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
13	12	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝐴:ℕ₀⟶ℤ)
14		xp1st 8018	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (1^st ‘𝑘) ∈ (0...𝑀))
15		elfznn0 13648	. . . . . . . . 9 ⊢ ((1^st ‘𝑘) ∈ (0...𝑀) → (1^st ‘𝑘) ∈ ℕ₀)
16	14, 15	syl 18	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (1^st ‘𝑘) ∈ ℕ₀)
17	16	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℕ₀)
18	13, 17	ffvelcdmd 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
19	18	zcnd 12701	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
20		reelprrecn 11192	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ℝ ∈ {ℝ, ℂ})
22		reopn 45900	. . . . . . . . 9 ⊢ ℝ ∈ (topGen‘ran (,))
23		tgioo4 24931	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
24	22, 23	eleqtri 2867	. . . . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → 𝑃 ∈ ℕ)
27		etransclem45.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
28	27	adantr 485	. . . . . . 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 8019	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ (0...𝑅))
31		elfznn0 13648	. . . . . . . . 9 ⊢ ((2^nd ‘𝑘) ∈ (0...𝑅) → (2^nd ‘𝑘) ∈ ℕ₀)
32	30, 31	syl 18	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ ℕ₀)
33	32	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (2^nd ‘𝑘) ∈ ℕ₀)
34	21, 25, 26, 28, 29, 33	etransclem33 46873	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
35	17	nn0red 12566	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℝ)
36	34, 35	ffvelcdmd 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
37	19, 36	mulcld 11229	. . . 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 12286	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
39	6, 11, 37, 38	fsumdivc 15837	. . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℂ)
41	38	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ≠ 0)
42	19, 36, 40, 41	divassd 12026	. . . . 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 46845	. . . . . . . 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 46851	. . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ (0...𝑀))
46	21, 25, 26, 28, 29, 33, 43, 44, 45, 35	etransclem37 46877	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))
47	10	nnzd 12617	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
48	47	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℤ)
49	17	nn0zd 12616	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℤ)
50	21, 25, 26, 28, 29, 33, 35, 49	etransclem42 46882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
51		dvdsval2 16313	. . . . . . . 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 1396	. . . . . . 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 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ)
54	18, 53	zmulcld 12706	. . . . 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 2869	. . . 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 15786	. . 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 2869	. 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 2873	1 ⊢ (𝜑 → 𝐾 ∈ ℤ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 {crab 3423 ifcif 4492 {cpr 4596 class class class wbr 5113 ↦ cmpt 5196 × cxp 5660 ran crn 5663 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 1^st c1st 7984 2^nd c2nd 7985 ↑_m cmap 8824 Fincfn 8943 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 · cmul 11105 − cmin 11441 / cdiv 11871 ℕcn 12233 ℕ₀cn0 12504 ℤcz 12591 (,)cioo 13372 ...cfz 13535 ↑cexp 14097 !cfa 14309 Σcsu 15737 ∏cprod 15957 ∥ cdvds 16310 ↾_t crest 17473 TopOpenctopn 17474 topGenctg 17490 ℂ_fldccnfld 21491 D^𝑛 cdvn 25992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-prod 15958 df-dvds 16311 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-limc 25994 df-dv 25995 df-dvn 25996

This theorem is referenced by: etransclem47 46887

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