Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > psercn2 | Structured version Visualization version GIF version |
Description: Since by pserulm 24937 the series converges uniformly, it is also continuous by ulmcn 24914. (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 5942 | . . . . . . . 8 ⊢ (◡abs “ (0[,]𝑀)) ⊆ dom abs | |
5 | absf 14685 | . . . . . . . . 9 ⊢ abs:ℂ⟶ℝ | |
6 | 5 | fdmi 6517 | . . . . . . . 8 ⊢ dom abs = ℂ |
7 | 4, 6 | sseqtri 4000 | . . . . . . 7 ⊢ (◡abs “ (0[,]𝑀)) ⊆ ℂ |
8 | 3, 7 | sstrdi 3976 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑆 ⊆ ℂ) |
10 | 9 | resmptd 5901 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
11 | simplr 765 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ) | |
12 | elfznn0 12988 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0) | |
13 | 12 | adantl 482 | . . . . . . . . 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 24926 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘))) |
16 | 11, 13, 15 | syl2anc 584 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘))) |
17 | simpr 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
18 | 17, 1 | eleqtrdi 2920 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ (ℤ≥‘0)) |
19 | 18 | adantr 481 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) → 𝑖 ∈ (ℤ≥‘0)) |
20 | pserf.a | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
21 | 20 | adantr 481 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ) |
22 | 21 | ffvelrnda 6843 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
23 | 22 | adantlr 711 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
24 | expcl 13435 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) | |
25 | 24 | adantll 710 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) |
26 | 23, 25 | mulcld 10649 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
27 | 12, 26 | sylan2 592 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
28 | 16, 19, 27 | fsumser 15075 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖)) |
29 | 28 | mpteq2dva 5152 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
30 | eqid 2818 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
31 | 30 | cnfldtopon 23318 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
32 | 31 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
33 | fzfid 13329 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (0...𝑖) ∈ Fin) | |
34 | 31 | a1i 11 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
35 | ffvelrn 6841 | . . . . . . . . . . 11 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) | |
36 | 21, 12, 35 | syl2an 595 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝐴‘𝑘) ∈ ℂ) |
37 | 34, 34, 36 | cnmptc 22198 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ (𝐴‘𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
38 | 12 | adantl 482 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0) |
39 | 30 | expcn 23407 | . . . . . . . . . 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 23402 | . . . . . . . . . 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 22202 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
44 | 30, 32, 33, 43 | fsumcn 23405 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
45 | 30 | cncfcn1 23445 | . . . . . . 7 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
46 | 44, 45 | eleqtrrdi 2921 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ (ℂ–cn→ℂ)) |
47 | 29, 46 | eqeltrrd 2911 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ)) |
48 | rescncf 23432 | . . . . 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 2911 | . . 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 6870 | . 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 24937 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑢‘𝑆)𝐹) |
58 | 1, 2, 52, 57 | ulmcn 24914 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 ◡ccnv 5547 dom cdm 5548 ↾ cres 5550 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 supcsup 8892 ℂcc 10523 ℝcr 10524 0cc0 10525 + caddc 10528 · cmul 10530 ℝ*cxr 10662 < clt 10663 ℕ0cn0 11885 ℤ≥cuz 12231 [,]cicc 12729 ...cfz 12880 seqcseq 13357 ↑cexp 13417 abscabs 14581 ⇝ cli 14829 Σcsu 15030 TopOpenctopn 16683 ℂfldccnfld 20473 TopOnctopon 21446 Cn ccn 21760 ×t ctx 22096 –cn→ccncf 23411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 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 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cn 21763 df-cnp 21764 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-ulm 24892 |
This theorem is referenced by: psercn 24941 |
Copyright terms: Public domain | W3C validator |