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| Mirrors > Home > MPE Home > Th. List > pserdv2 | Structured version Visualization version GIF version | ||
| Description: The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-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*, < ) | 
| psercn.s | ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) | 
| psercn.m | ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) | 
| pserdv.b | ⊢ 𝐵 = (0(ball‘(abs ∘ − ))(((abs‘𝑎) + 𝑀) / 2)) | 
| Ref | Expression | 
|---|---|
| pserdv2 | ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pserf.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
| 2 | pserf.f | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) | |
| 3 | pserf.a | . . 3 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
| 4 | pserf.r | . . 3 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
| 5 | psercn.s | . . 3 ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) | |
| 6 | psercn.m | . . 3 ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) | |
| 7 | pserdv.b | . . 3 ⊢ 𝐵 = (0(ball‘(abs ∘ − ))(((abs‘𝑎) + 𝑀) / 2)) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | pserdv 26473 | . 2 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑚 ∈ ℕ0 (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚)))) | 
| 9 | nn0uz 12920 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
| 10 | nnuz 12921 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
| 11 | 1e0p1 12775 | . . . . . . 7 ⊢ 1 = (0 + 1) | |
| 12 | 11 | fveq2i 6909 | . . . . . 6 ⊢ (ℤ≥‘1) = (ℤ≥‘(0 + 1)) | 
| 13 | 10, 12 | eqtri 2765 | . . . . 5 ⊢ ℕ = (ℤ≥‘(0 + 1)) | 
| 14 | id 22 | . . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → 𝑘 = (1 + 𝑚)) | |
| 15 | fveq2 6906 | . . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝐴‘𝑘) = (𝐴‘(1 + 𝑚))) | |
| 16 | 14, 15 | oveq12d 7449 | . . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 · (𝐴‘𝑘)) = ((1 + 𝑚) · (𝐴‘(1 + 𝑚)))) | 
| 17 | oveq1 7438 | . . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 − 1) = ((1 + 𝑚) − 1)) | |
| 18 | 17 | oveq2d 7447 | . . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑦↑(𝑘 − 1)) = (𝑦↑((1 + 𝑚) − 1))) | 
| 19 | 16, 18 | oveq12d 7449 | . . . . 5 ⊢ (𝑘 = (1 + 𝑚) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) = (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1)))) | 
| 20 | 1zzd 12648 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ ℤ) | |
| 21 | 0zd 12625 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 0 ∈ ℤ) | |
| 22 | nncn 12274 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ) | |
| 23 | 22 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ) | 
| 24 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐴:ℕ0⟶ℂ) | 
| 25 | nnnn0 12533 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
| 26 | ffvelcdm 7101 | . . . . . . . 8 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) | |
| 27 | 24, 25, 26 | syl2an 596 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℂ) | 
| 28 | 23, 27 | mulcld 11281 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑘 · (𝐴‘𝑘)) ∈ ℂ) | 
| 29 | cnvimass 6100 | . . . . . . . . . . 11 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs | |
| 30 | absf 15376 | . . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ | |
| 31 | 30 | fdmi 6747 | . . . . . . . . . . 11 ⊢ dom abs = ℂ | 
| 32 | 29, 31 | sseqtri 4032 | . . . . . . . . . 10 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ | 
| 33 | 5, 32 | eqsstri 4030 | . . . . . . . . 9 ⊢ 𝑆 ⊆ ℂ | 
| 34 | 33 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | 
| 35 | 34 | sselda 3983 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ) | 
| 36 | nnm1nn0 12567 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ0) | |
| 37 | expcl 14120 | . . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑘 − 1) ∈ ℕ0) → (𝑦↑(𝑘 − 1)) ∈ ℂ) | |
| 38 | 35, 36, 37 | syl2an 596 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑦↑(𝑘 − 1)) ∈ ℂ) | 
| 39 | 28, 38 | mulcld 11281 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) ∈ ℂ) | 
| 40 | 9, 13, 19, 20, 21, 39 | isumshft 15875 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) = Σ𝑚 ∈ ℕ0 (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1)))) | 
| 41 | ax-1cn 11213 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
| 42 | nn0cn 12536 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ) | |
| 43 | 42 | adantl 481 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ) | 
| 44 | addcom 11447 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (1 + 𝑚) = (𝑚 + 1)) | |
| 45 | 41, 43, 44 | sylancr 587 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ0) → (1 + 𝑚) = (𝑚 + 1)) | 
| 46 | 45 | fveq2d 6910 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ0) → (𝐴‘(1 + 𝑚)) = (𝐴‘(𝑚 + 1))) | 
| 47 | 45, 46 | oveq12d 7449 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ0) → ((1 + 𝑚) · (𝐴‘(1 + 𝑚))) = ((𝑚 + 1) · (𝐴‘(𝑚 + 1)))) | 
| 48 | pncan2 11515 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚) | |
| 49 | 41, 43, 48 | sylancr 587 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ0) → ((1 + 𝑚) − 1) = 𝑚) | 
| 50 | 49 | oveq2d 7447 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ0) → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚)) | 
| 51 | 47, 50 | oveq12d 7449 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ0) → (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1))) = (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚))) | 
| 52 | 51 | sumeq2dv 15738 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → Σ𝑚 ∈ ℕ0 (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1))) = Σ𝑚 ∈ ℕ0 (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚))) | 
| 53 | 40, 52 | eqtr2d 2778 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → Σ𝑚 ∈ ℕ0 (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚)) = Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1)))) | 
| 54 | 53 | mpteq2dva 5242 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑆 ↦ Σ𝑚 ∈ ℕ0 (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚))) = (𝑦 ∈ 𝑆 ↦ Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))))) | 
| 55 | 8, 54 | eqtrd 2777 | 1 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 ifcif 4525 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 “ cima 5688 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℝ*cxr 11294 < clt 11295 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ℤ≥cuz 12878 [,)cico 13389 seqcseq 14042 ↑cexp 14102 abscabs 15273 ⇝ cli 15520 Σcsu 15722 ballcbl 21351 D cdv 25898 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-ulm 26420 | 
| This theorem is referenced by: logtayl 26702 binomcxplemdvsum 44374 | 
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