etransclem45 - Mathbox for Glauco Siliprandi

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Theorem etransclem45 46637

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 13907	. . . . . 6 ⊢ (0...𝑀) ∈ Fin
3		fzfi 13907	. . . . . 6 ⊢ (0...𝑅) ∈ Fin
4		xpfi 9232	. . . . . 6 ⊢ (((0...𝑀) ∈ Fin ∧ (0...𝑅) ∈ Fin) → ((0...𝑀) × (0...𝑅)) ∈ Fin)
5	2, 3, 4	mp2an 693	. . . . 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 12454	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
10	9	faccld 14219	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
11	10	nncnd 12173	. . . 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 7975	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (1^st ‘𝑘) ∈ (0...𝑀))
15		elfznn0 13548	. . . . . . . . 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 7039	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
19	18	zcnd 12609	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
20		reelprrecn 11130	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ℝ ∈ {ℝ, ℂ})
22		reopn 45651	. . . . . . . . 9 ⊢ ℝ ∈ (topGen‘ran (,))
23		tgioo4 24761	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
24	22, 23	eleqtri 2835	. . . . . . . 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 7976	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...𝑅)) → (2^nd ‘𝑘) ∈ (0...𝑅))
31		elfznn0 13548	. . . . . . . . 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 46625	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
35	17	nn0red 12475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℝ)
36	34, 35	ffvelcdmd 7039	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
37	19, 36	mulcld 11164	. . . 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 12207	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
39	6, 11, 37, 38	fsumdivc 15721	. . 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 11964	. . . . 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 46597	. . . . . . . 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 46603	. . . . . . . 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 46629	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))
47	10	nnzd 12526	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
48	47	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (!‘(𝑃 − 1)) ∈ ℤ)
49	17	nn0zd 12525	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (1^st ‘𝑘) ∈ ℤ)
50	21, 25, 26, 28, 29, 33, 35, 49	etransclem42 46634	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...𝑅))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
51		dvdsval2 16194	. . . . . . . 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 1374	. . . . . . 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 12614	. . . . 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 2837	. . . 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 15670	. . 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 2837	. 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 2841	1 ⊢ (𝜑 → 𝐾 ∈ ℤ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {crab 3401 ifcif 4481 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 × cxp 5630 ran crn 5633 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 ↑_m cmap 8775 Fincfn 8895 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 − cmin 11376 / cdiv 11806 ℕcn 12157 ℕ₀cn0 12413 ℤcz 12500 (,)cioo 13273 ...cfz 13435 ↑cexp 13996 !cfa 14208 Σcsu 15621 ∏cprod 15838 ∥ cdvds 16191 ↾_t crest 17352 TopOpenctopn 17353 topGenctg 17369 ℂ_fldccnfld 21321 D^𝑛 cdvn 25833

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-prod 15839 df-dvds 16192 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836 df-dvn 25837

This theorem is referenced by: etransclem47 46639

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