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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem39 | Structured version Visualization version GIF version |
Description: 𝐺 is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem39.p | ⊢ (𝜑 → 𝑃 ∈ ℕ) |
etransclem39.m | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
etransclem39.f | ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))) |
etransclem39.g | ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥)) |
Ref | Expression |
---|---|
etransclem39 | ⊢ (𝜑 → 𝐺:ℝ⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13832 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (0...𝑅) ∈ Fin) | |
2 | reelprrecn 11101 | . . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ} | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ℝ ∈ {ℝ, ℂ}) |
4 | reopn 43421 | . . . . . . . 8 ⊢ ℝ ∈ (topGen‘ran (,)) | |
5 | eqid 2737 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
6 | 5 | tgioo2 24117 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
7 | 4, 6 | eleqtri 2836 | . . . . . . 7 ⊢ ℝ ∈ ((TopOpen‘ℂfld) ↾t ℝ) |
8 | 7 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ℝ ∈ ((TopOpen‘ℂfld) ↾t ℝ)) |
9 | etransclem39.p | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) | |
10 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑃 ∈ ℕ) |
11 | etransclem39.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
12 | 11 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑀 ∈ ℕ0) |
13 | etransclem39.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))) | |
14 | elfznn0 13488 | . . . . . . 7 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ0) | |
15 | 14 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ0) |
16 | 3, 8, 10, 12, 13, 15 | etransclem33 44402 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D𝑛 𝐹)‘𝑖):ℝ⟶ℂ) |
17 | 16 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ (0...𝑅)) → ((ℝ D𝑛 𝐹)‘𝑖):ℝ⟶ℂ) |
18 | simplr 767 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ (0...𝑅)) → 𝑥 ∈ ℝ) | |
19 | 17, 18 | ffvelcdmd 7032 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ (0...𝑅)) → (((ℝ D𝑛 𝐹)‘𝑖)‘𝑥) ∈ ℂ) |
20 | 1, 19 | fsumcl 15577 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥) ∈ ℂ) |
21 | etransclem39.g | . 2 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥)) | |
22 | 20, 21 | fmptd 7058 | 1 ⊢ (𝜑 → 𝐺:ℝ⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cpr 4586 ↦ cmpt 5186 ran crn 5632 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 · cmul 11014 − cmin 11343 ℕcn 12111 ℕ0cn0 12371 (,)cioo 13218 ...cfz 13378 ↑cexp 13921 Σcsu 15529 ∏cprod 15747 ↾t crest 17261 TopOpenctopn 17262 topGenctg 17278 ℂfldccnfld 20748 D𝑛 cdvn 25179 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ioo 13222 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-fac 14127 df-bc 14156 df-hash 14184 df-cj 14943 df-re 14944 df-im 14945 df-sqrt 15079 df-abs 15080 df-clim 15329 df-sum 15530 df-prod 15748 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-starv 17107 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-unif 17115 df-hom 17116 df-cco 17117 df-rest 17263 df-topn 17264 df-0g 17282 df-gsum 17283 df-topgen 17284 df-pt 17285 df-prds 17288 df-xrs 17343 df-qtop 17348 df-imas 17349 df-xps 17351 df-mre 17425 df-mrc 17426 df-acs 17428 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-submnd 18561 df-mulg 18831 df-cntz 19055 df-cmn 19522 df-psmet 20740 df-xmet 20741 df-met 20742 df-bl 20743 df-mopn 20744 df-fbas 20745 df-fg 20746 df-cnfld 20749 df-top 22194 df-topon 22211 df-topsp 22233 df-bases 22247 df-cld 22321 df-ntr 22322 df-cls 22323 df-nei 22400 df-lp 22438 df-perf 22439 df-cn 22529 df-cnp 22530 df-haus 22617 df-tx 22864 df-hmeo 23057 df-fil 23148 df-fm 23240 df-flim 23241 df-flf 23242 df-xms 23624 df-ms 23625 df-tms 23626 df-cncf 24192 df-limc 25181 df-dv 25182 df-dvn 25183 |
This theorem is referenced by: etransclem46 44415 |
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