Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem36 | Structured version Visualization version GIF version |
Description: The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem36.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
etransclem36.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
etransclem36.p | ⊢ (𝜑 → 𝑃 ∈ ℕ) |
etransclem36.m | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
etransclem36.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))) |
etransclem36.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
etransclem36.h | ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) |
etransclem36.jx | ⊢ (𝜑 → 𝐽 ∈ 𝑋) |
etransclem36.jz | ⊢ (𝜑 → 𝐽 ∈ ℤ) |
etransclem36.10 | ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) |
Ref | Expression |
---|---|
etransclem36 | ⊢ (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem36.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | etransclem36.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
3 | etransclem36.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ) | |
4 | etransclem36.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
5 | etransclem36.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))) | |
6 | etransclem36.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
7 | etransclem36.h | . . 3 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) | |
8 | etransclem36.10 | . . 3 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) | |
9 | etransclem36.jx | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑋) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | etransclem31 43332 | . 2 ⊢ (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))))) |
11 | 8, 6 | etransclem16 43317 | . . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin) |
12 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑃 ∈ ℕ) |
13 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ ℕ0) |
14 | 6 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 ∈ ℕ0) |
15 | etransclem36.jz | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) | |
16 | 15 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ ℤ) |
17 | etransclem11 43312 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) | |
18 | etransclem11 43312 | . . . . 5 ⊢ (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) = (𝑛 ∈ ℕ0 ↦ {𝑒 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑒‘𝑗) = 𝑛}) | |
19 | 8, 17, 18 | 3eqtri 2765 | . . . 4 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑒 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑒‘𝑗) = 𝑛}) |
20 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁)) | |
21 | 12, 13, 14, 16, 19, 20 | etransclem26 43327 | . . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))) ∈ ℤ) |
22 | 11, 21 | fsumzcl 15178 | . 2 ⊢ (𝜑 → Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))) ∈ ℤ) |
23 | 10, 22 | eqeltrd 2833 | 1 ⊢ (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {crab 3057 ifcif 4411 {cpr 4515 class class class wbr 5027 ↦ cmpt 5107 ‘cfv 6333 (class class class)co 7164 ↑m cmap 8430 ℂcc 10606 ℝcr 10607 0cc0 10608 1c1 10609 · cmul 10613 < clt 10746 − cmin 10941 / cdiv 11368 ℕcn 11709 ℕ0cn0 11969 ℤcz 12055 ...cfz 12974 ↑cexp 13514 !cfa 13718 Σcsu 15128 ∏cprod 15344 ↾t crest 16790 TopOpenctopn 16791 ℂfldccnfld 20210 D𝑛 cdvn 24608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ico 12820 df-icc 12821 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-fac 13719 df-bc 13748 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-clim 14928 df-sum 15129 df-prod 15345 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-rest 16792 df-topn 16793 df-0g 16811 df-gsum 16812 df-topgen 16813 df-pt 16814 df-prds 16817 df-xrs 16871 df-qtop 16876 df-imas 16877 df-xps 16879 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-mulg 18336 df-cntz 18558 df-cmn 19019 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-fbas 20207 df-fg 20208 df-cnfld 20211 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-nei 21842 df-lp 21880 df-perf 21881 df-cn 21971 df-cnp 21972 df-haus 22059 df-tx 22306 df-hmeo 22499 df-fil 22590 df-fm 22682 df-flim 22683 df-flf 22684 df-xms 23066 df-ms 23067 df-tms 23068 df-cncf 23623 df-limc 24610 df-dv 24611 df-dvn 24612 |
This theorem is referenced by: etransclem42 43343 |
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