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Mirrors > Home > MPE Home > Th. List > psercn2 | Structured version Visualization version GIF version |
Description: Since by pserulm 24176 the series converges uniformly, it is also continuous by ulmcn 24153. (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 11722 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11389 | . 2 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | pserulm.y | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀))) | |
4 | cnvimass 5485 | . . . . . . . 8 ⊢ (◡abs “ (0[,]𝑀)) ⊆ dom abs | |
5 | absf 14077 | . . . . . . . . 9 ⊢ abs:ℂ⟶ℝ | |
6 | 5 | fdmi 6052 | . . . . . . . 8 ⊢ dom abs = ℂ |
7 | 4, 6 | sseqtri 3637 | . . . . . . 7 ⊢ (◡abs “ (0[,]𝑀)) ⊆ ℂ |
8 | 3, 7 | syl6ss 3615 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑆 ⊆ ℂ) |
10 | 9 | resmptd 5452 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
11 | simplr 792 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ) | |
12 | elfznn0 12433 | . . . . . . . . . 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 24165 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘))) |
16 | 11, 13, 15 | syl2anc 693 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘))) |
17 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
18 | 17, 1 | syl6eleq 2711 | . . . . . . . . 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 6359 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
23 | 22 | adantlr 751 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
24 | expcl 12878 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) | |
25 | 24 | adantll 750 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) |
26 | 23, 25 | mulcld 10060 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
27 | 12, 26 | sylan2 491 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
28 | 16, 19, 27 | fsumser 14461 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖)) |
29 | 28 | mpteq2dva 4744 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
30 | eqid 2622 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
31 | 30 | cnfldtopon 22586 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
32 | 31 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
33 | fzfid 12772 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (0...𝑖) ∈ Fin) | |
34 | 31 | a1i 11 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
35 | ffvelrn 6357 | . . . . . . . . . . 11 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) | |
36 | 21, 12, 35 | syl2an 494 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝐴‘𝑘) ∈ ℂ) |
37 | 34, 34, 36 | cnmptc 21465 | . . . . . . . . 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 22675 | . . . . . . . . . 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 22670 | . . . . . . . . . 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 21469 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
44 | 30, 32, 33, 43 | fsumcn 22673 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
45 | 30 | cncfcn1 22713 | . . . . . . 7 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
46 | 44, 45 | syl6eleqr 2712 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ (ℂ–cn→ℂ)) |
47 | 29, 46 | eqeltrrd 2702 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ)) |
48 | rescncf 22700 | . . . . 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 2702 | . . 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 6385 | . 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 24176 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑢‘𝑆)𝐹) |
58 | 1, 2, 52, 57 | ulmcn 24153 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ◡ccnv 5113 dom cdm 5114 ↾ cres 5116 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 · cmul 9941 ℝ*cxr 10073 < clt 10074 ℕ0cn0 11292 ℤ≥cuz 11687 [,]cicc 12178 ...cfz 12326 seqcseq 12801 ↑cexp 12860 abscabs 13974 ⇝ cli 14215 Σcsu 14416 TopOpenctopn 16082 ℂfldccnfld 19746 TopOnctopon 20715 Cn ccn 21028 ×t ctx 21363 –cn→ccncf 22679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ulm 24131 |
This theorem is referenced by: psercn 24180 |
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