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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem43 | Structured version Visualization version GIF version |
Description: 𝐺 is a continuous function. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem43.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
etransclem43.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
etransclem43.p | ⊢ (𝜑 → 𝑃 ∈ ℕ) |
etransclem43.m | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
etransclem43.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))) |
etransclem43.g | ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥)) |
Ref | Expression |
---|---|
etransclem43 | ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem43.g | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥)) | |
2 | etransclem43.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
3 | etransclem43.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
4 | 2, 3 | dvdmsscn 43968 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
5 | fzfid 13808 | . . 3 ⊢ (𝜑 → (0...𝑅) ∈ Fin) | |
6 | 2 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑆 ∈ {ℝ, ℂ}) |
7 | 3 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
8 | etransclem43.p | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) | |
9 | 8 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑃 ∈ ℕ) |
10 | etransclem43.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
11 | 10 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑀 ∈ ℕ0) |
12 | etransclem43.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))) | |
13 | elfznn0 13464 | . . . . . . 7 ⊢ (𝑖 ∈ (0...𝑅) → 𝑖 ∈ ℕ0) | |
14 | 13 | adantl 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → 𝑖 ∈ ℕ0) |
15 | 6, 7, 9, 11, 12, 14 | etransclem33 44299 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D𝑛 𝐹)‘𝑖):𝑋⟶ℂ) |
16 | 15 | feqmptd 6906 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D𝑛 𝐹)‘𝑖) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥))) |
17 | 6, 7, 9, 11, 12, 14 | etransclem40 44306 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → ((𝑆 D𝑛 𝐹)‘𝑖) ∈ (𝑋–cn→ℂ)) |
18 | 16, 17 | eqeltrrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑅)) → (𝑥 ∈ 𝑋 ↦ (((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ)) |
19 | 4, 5, 18 | fsumcncf 43910 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥)) ∈ (𝑋–cn→ℂ)) |
20 | 1, 19 | eqeltrid 2843 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cpr 4587 ↦ cmpt 5187 ‘cfv 6492 (class class class)co 7350 ℂcc 10983 ℝcr 10984 0cc0 10985 1c1 10986 · cmul 10990 − cmin 11319 ℕcn 12087 ℕ0cn0 12347 ...cfz 13354 ↑cexp 13897 Σcsu 15506 ∏cprod 15724 ↾t crest 17238 TopOpenctopn 17239 ℂfldccnfld 20725 –cn→ccncf 24167 D𝑛 cdvn 25156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 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 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8582 df-map 8701 df-pm 8702 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-fi 9281 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-q 12804 df-rp 12846 df-xneg 12963 df-xadd 12964 df-xmul 12965 df-ico 13200 df-icc 13201 df-fz 13355 df-fzo 13498 df-seq 13837 df-exp 13898 df-fac 14103 df-bc 14132 df-hash 14160 df-cj 14919 df-re 14920 df-im 14921 df-sqrt 15055 df-abs 15056 df-clim 15306 df-sum 15507 df-prod 15725 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-starv 17084 df-sca 17085 df-vsca 17086 df-ip 17087 df-tset 17088 df-ple 17089 df-ds 17091 df-unif 17092 df-hom 17093 df-cco 17094 df-rest 17240 df-topn 17241 df-0g 17259 df-gsum 17260 df-topgen 17261 df-pt 17262 df-prds 17265 df-xrs 17320 df-qtop 17325 df-imas 17326 df-xps 17328 df-mre 17402 df-mrc 17403 df-acs 17405 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-submnd 18538 df-mulg 18808 df-cntz 19032 df-cmn 19499 df-psmet 20717 df-xmet 20718 df-met 20719 df-bl 20720 df-mopn 20721 df-fbas 20722 df-fg 20723 df-cnfld 20726 df-top 22171 df-topon 22188 df-topsp 22210 df-bases 22224 df-cld 22298 df-ntr 22299 df-cls 22300 df-nei 22377 df-lp 22415 df-perf 22416 df-cn 22506 df-cnp 22507 df-haus 22594 df-tx 22841 df-hmeo 23034 df-fil 23125 df-fm 23217 df-flim 23218 df-flf 23219 df-xms 23601 df-ms 23602 df-tms 23603 df-cncf 24169 df-limc 25158 df-dv 25159 df-dvn 25160 |
This theorem is referenced by: etransclem46 44312 |
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