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Mirrors > Home > MPE Home > Th. List > psercn2 | Structured version Visualization version GIF version |
Description: Since by pserulm 25017 the series converges uniformly, it is also continuous by ulmcn 24994. (Contributed by Mario Carneiro, 3-Mar-2015.) |
Ref | Expression |
---|---|
pserf.g | ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
pserf.f | ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) |
pserf.a | ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
pserf.r | ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) |
pserulm.h | ⊢ 𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
pserulm.m | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
pserulm.l | ⊢ (𝜑 → 𝑀 < 𝑅) |
pserulm.y | ⊢ (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀))) |
Ref | Expression |
---|---|
psercn2 | ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12268 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11981 | . 2 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | pserulm.y | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀))) | |
4 | cnvimass 5916 | . . . . . . . 8 ⊢ (◡abs “ (0[,]𝑀)) ⊆ dom abs | |
5 | absf 14689 | . . . . . . . . 9 ⊢ abs:ℂ⟶ℝ | |
6 | 5 | fdmi 6498 | . . . . . . . 8 ⊢ dom abs = ℂ |
7 | 4, 6 | sseqtri 3951 | . . . . . . 7 ⊢ (◡abs “ (0[,]𝑀)) ⊆ ℂ |
8 | 3, 7 | sstrdi 3927 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑆 ⊆ ℂ) |
10 | 9 | resmptd 5875 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
11 | simplr 768 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ) | |
12 | elfznn0 12995 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0) | |
13 | 12 | adantl 485 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0) |
14 | pserf.g | . . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
15 | 14 | pserval2 25006 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘))) |
16 | 11, 13, 15 | syl2anc 587 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘))) |
17 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
18 | 17, 1 | eleqtrdi 2900 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ (ℤ≥‘0)) |
19 | 18 | adantr 484 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) → 𝑖 ∈ (ℤ≥‘0)) |
20 | pserf.a | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
21 | 20 | adantr 484 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ) |
22 | 21 | ffvelrnda 6828 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
23 | 22 | adantlr 714 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
24 | expcl 13443 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) | |
25 | 24 | adantll 713 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) |
26 | 23, 25 | mulcld 10650 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
27 | 12, 26 | sylan2 595 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
28 | 16, 19, 27 | fsumser 15079 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖)) |
29 | 28 | mpteq2dva 5125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
30 | eqid 2798 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
31 | 30 | cnfldtopon 23388 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
32 | 31 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
33 | fzfid 13336 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (0...𝑖) ∈ Fin) | |
34 | 31 | a1i 11 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
35 | ffvelrn 6826 | . . . . . . . . . . 11 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) | |
36 | 21, 12, 35 | syl2an 598 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝐴‘𝑘) ∈ ℂ) |
37 | 34, 34, 36 | cnmptc 22267 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ (𝐴‘𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
38 | 12 | adantl 485 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0) |
39 | 30 | expcn 23477 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
40 | 38, 39 | syl 17 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
41 | 30 | mulcn 23472 | . . . . . . . . . 10 ⊢ · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
42 | 41 | a1i 11 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
43 | 34, 37, 40, 42 | cnmpt12f 22271 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
44 | 30, 32, 33, 43 | fsumcn 23475 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
45 | 30 | cncfcn1 23516 | . . . . . . 7 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
46 | 44, 45 | eleqtrrdi 2901 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ (ℂ–cn→ℂ)) |
47 | 29, 46 | eqeltrrd 2891 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ)) |
48 | rescncf 23502 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) ∈ (𝑆–cn→ℂ))) | |
49 | 9, 47, 48 | sylc 65 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) ∈ (𝑆–cn→ℂ)) |
50 | 10, 49 | eqeltrrd 2891 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (𝑆–cn→ℂ)) |
51 | pserulm.h | . . 3 ⊢ 𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) | |
52 | 50, 51 | fmptd 6855 | . 2 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝑆–cn→ℂ)) |
53 | pserf.f | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) | |
54 | pserf.r | . . 3 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
55 | pserulm.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
56 | pserulm.l | . . 3 ⊢ (𝜑 → 𝑀 < 𝑅) | |
57 | 14, 53, 20, 54, 51, 55, 56, 3 | pserulm 25017 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑢‘𝑆)𝐹) |
58 | 1, 2, 52, 57 | ulmcn 24994 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ◡ccnv 5518 dom cdm 5519 ↾ cres 5521 “ cima 5522 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 ℝ*cxr 10663 < clt 10664 ℕ0cn0 11885 ℤ≥cuz 12231 [,]cicc 12729 ...cfz 12885 seqcseq 13364 ↑cexp 13425 abscabs 14585 ⇝ cli 14833 Σcsu 15034 TopOpenctopn 16687 ℂfldccnfld 20091 TopOnctopon 21515 Cn ccn 21829 ×t ctx 22165 –cn→ccncf 23481 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-ulm 24972 |
This theorem is referenced by: psercn 25021 |
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