pserdv2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pserdv2		Structured version Visualization version GIF version

Theorem pserdv2 26551

Description: The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
pserdv2	⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1)))))

Distinct variable groups: 𝑗,𝑎,𝑘,𝑛,𝑟,𝑥,𝑦,𝐴 𝑗,𝑀,𝑘,𝑦 𝐵,𝑗,𝑘,𝑥,𝑦 𝑗,𝐺,𝑘,𝑟,𝑦 𝑆,𝑎,𝑗,𝑘,𝑦 𝐹,𝑎 𝜑,𝑎,𝑗,𝑘,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑛,𝑟) 𝐵(𝑛,𝑟,𝑎) 𝑅(𝑥,𝑦,𝑗,𝑘,𝑛,𝑟,𝑎) 𝑆(𝑥,𝑛,𝑟) 𝐹(𝑥,𝑦,𝑗,𝑘,𝑛,𝑟) 𝐺(𝑥,𝑛,𝑎) 𝑀(𝑥,𝑛,𝑟,𝑎)

Proof of Theorem pserdv2 Dummy variable 𝑚 is distinct from all other variables.

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 26550	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑚 ∈ ℕ₀ (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚))))
9		nn0uz 12891	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
10		nnuz 12892	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
11		1e0p1 12749	. . . . . . 7 ⊢ 1 = (0 + 1)
12	11	fveq2i 6874	. . . . . 6 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
13	10, 12	eqtri 2788	. . . . 5 ⊢ ℕ = (ℤ_≥‘(0 + 1))
14		id 23	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → 𝑘 = (1 + 𝑚))
15		fveq2 6871	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝐴‘𝑘) = (𝐴‘(1 + 𝑚)))
16	14, 15	oveq12d 7418	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 · (𝐴‘𝑘)) = ((1 + 𝑚) · (𝐴‘(1 + 𝑚))))
17		oveq1 7407	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 − 1) = ((1 + 𝑚) − 1))
18	17	oveq2d 7416	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑦↑(𝑘 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
19	16, 18	oveq12d 7418	. . . . 5 ⊢ (𝑘 = (1 + 𝑚) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) = (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1))))
20		1zzd 12616	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ ℤ)
21		0zd 12594	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 0 ∈ ℤ)
22		nncn 12232	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
23	22	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
24	3	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐴:ℕ₀⟶ℂ)
25		nnnn0 12502	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
26		ffvelcdm 7066	. . . . . . . 8 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
27	24, 25, 26	syl2an 607	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℂ)
28	23, 27	mulcld 11217	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑘 · (𝐴‘𝑘)) ∈ ℂ)
29		cnvimass 6075	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
30		absf 15379	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
31	30	fdmi 6707	. . . . . . . . . . 11 ⊢ dom abs = ℂ
32	29, 31	sseqtri 3987	. . . . . . . . . 10 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
33	5, 32	eqsstri 3985	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℂ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
35	34	sselda 3939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
36		nnm1nn0 12536	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
37		expcl 14106	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑘 − 1) ∈ ℕ₀) → (𝑦↑(𝑘 − 1)) ∈ ℂ)
38	35, 36, 37	syl2an 607	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑦↑(𝑘 − 1)) ∈ ℂ)
39	28, 38	mulcld 11217	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) ∈ ℂ)
40	9, 13, 19, 20, 21, 39	isumshft 15883	. . . 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 11146	. . . . . . . 8 ⊢ 1 ∈ ℂ
42		nn0cn 12505	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
43	42	adantl 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
44		addcom 11384	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (1 + 𝑚) = (𝑚 + 1))
45	41, 43, 44	sylancr 598	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (1 + 𝑚) = (𝑚 + 1))
46	45	fveq2d 6875	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝐴‘(1 + 𝑚)) = (𝐴‘(𝑚 + 1)))
47	45, 46	oveq12d 7418	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((1 + 𝑚) · (𝐴‘(1 + 𝑚))) = ((𝑚 + 1) · (𝐴‘(𝑚 + 1))))
48		pncan2 11452	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
49	41, 43, 48	sylancr 598	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((1 + 𝑚) − 1) = 𝑚)
50	49	oveq2d 7416	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚))
51	47, 50	oveq12d 7418	. . . . 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 15743	. . . 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 2801	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → Σ𝑚 ∈ ℕ₀ (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚)) = Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))))
54	53	mpteq2dva 5198	. 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 2800	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 400 = wceq 1563 ∈ wcel 2145 {crab 3417 ⊆ wss 3907 ifcif 4483 ↦ cmpt 5186 ^◡ccnv 5651 dom cdm 5652 “ cima 5655 ∘ ccom 5656 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 supcsup 9388 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 ℝ^*cxr 11230 < clt 11231 − cmin 11429 / cdiv 11859 ℕcn 12224 2c2 12286 ℕ₀cn0 12495 ℤ_≥cuz 12853 [,)cico 13365 seqcseq 14028 ↑cexp 14088 abscabs 15275 ⇝ cli 15525 Σcsu 15727 ballcbl 21469 D cdv 25983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-seq 14029 df-exp 14089 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-cmp 23505 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-limc 25986 df-dv 25987 df-ulm 26498

This theorem is referenced by: logtayl 26783 binomcxplemdvsum 44929

W3C validator