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Mirrors > Home > MPE Home > Th. List > psercn2 | Structured version Visualization version GIF version |
Description: Since by pserulm 24728 the series converges uniformly, it is also continuous by ulmcn 24705. (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 12092 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11803 | . 2 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | pserulm.y | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀))) | |
4 | cnvimass 5786 | . . . . . . . 8 ⊢ (◡abs “ (0[,]𝑀)) ⊆ dom abs | |
5 | absf 14556 | . . . . . . . . 9 ⊢ abs:ℂ⟶ℝ | |
6 | 5 | fdmi 6351 | . . . . . . . 8 ⊢ dom abs = ℂ |
7 | 4, 6 | sseqtri 3886 | . . . . . . 7 ⊢ (◡abs “ (0[,]𝑀)) ⊆ ℂ |
8 | 3, 7 | syl6ss 3863 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 8 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑆 ⊆ ℂ) |
10 | 9 | resmptd 5750 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
11 | simplr 757 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → 𝑦 ∈ ℂ) | |
12 | elfznn0 12814 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0) | |
13 | 12 | adantl 474 | . . . . . . . . 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 24717 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑦)‘𝑘) = ((𝐴‘𝑘) · (𝑦↑𝑘))) |
16 | 11, 13, 15 | syl2anc 576 | . . . . . . . 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 2869 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ (ℤ≥‘0)) |
19 | 18 | adantr 473 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) → 𝑖 ∈ (ℤ≥‘0)) |
20 | pserf.a | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
21 | 20 | adantr 473 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ) |
22 | 21 | ffvelrnda 6674 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
23 | 22 | adantlr 703 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
24 | expcl 13260 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) | |
25 | 24 | adantll 702 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑦↑𝑘) ∈ ℂ) |
26 | 23, 25 | mulcld 10458 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
27 | 12, 26 | sylan2 584 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐴‘𝑘) · (𝑦↑𝑘)) ∈ ℂ) |
28 | 16, 19, 27 | fsumser 14945 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑦 ∈ ℂ) → Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘)) = (seq0( + , (𝐺‘𝑦))‘𝑖)) |
29 | 28 | mpteq2dva 5018 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖))) |
30 | eqid 2771 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
31 | 30 | cnfldtopon 23109 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
32 | 31 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
33 | fzfid 13154 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (0...𝑖) ∈ Fin) | |
34 | 31 | a1i 11 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
35 | ffvelrn 6672 | . . . . . . . . . . 11 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) | |
36 | 21, 12, 35 | syl2an 587 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝐴‘𝑘) ∈ ℂ) |
37 | 34, 34, 36 | cnmptc 21989 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ (𝐴‘𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
38 | 12 | adantl 474 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0) |
39 | 30 | expcn 23198 | . . . . . . . . . 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 23193 | . . . . . . . . . 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 21993 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑖)) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
44 | 30, 32, 33, 43 | fsumcn 23196 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
45 | 30 | cncfcn1 23236 | . . . . . . 7 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
46 | 44, 45 | syl6eleqr 2870 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑖)((𝐴‘𝑘) · (𝑦↑𝑘))) ∈ (ℂ–cn→ℂ)) |
47 | 29, 46 | eqeltrrd 2860 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)) ∈ (ℂ–cn→ℂ)) |
48 | rescncf 23223 | . . . . 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 2860 | . . 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 6699 | . 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 24728 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑢‘𝑆)𝐹) |
58 | 1, 2, 52, 57 | ulmcn 24705 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {crab 3085 ⊆ wss 3822 class class class wbr 4925 ↦ cmpt 5004 ◡ccnv 5402 dom cdm 5403 ↾ cres 5405 “ cima 5406 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 supcsup 8697 ℂcc 10331 ℝcr 10332 0cc0 10333 + caddc 10336 · cmul 10338 ℝ*cxr 10471 < clt 10472 ℕ0cn0 11705 ℤ≥cuz 12056 [,]cicc 12555 ...cfz 12706 seqcseq 13182 ↑cexp 13242 abscabs 14452 ⇝ cli 14700 Σcsu 14901 TopOpenctopn 16549 ℂfldccnfld 20262 TopOnctopon 21237 Cn ccn 21551 ×t ctx 21887 –cn→ccncf 23202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-limsup 14687 df-clim 14704 df-rlim 14705 df-sum 14902 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cn 21554 df-cnp 21555 df-tx 21889 df-hmeo 22082 df-xms 22648 df-ms 22649 df-tms 22650 df-cncf 23204 df-ulm 24683 |
This theorem is referenced by: psercn 24732 |
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