pserdv2 - Metamath Proof Explorer

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Theorem pserdv2 26420

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 26419	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑚 ∈ ℕ₀ (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚))))
9		nn0uz 12824	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
10		nnuz 12825	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
11		1e0p1 12684	. . . . . . 7 ⊢ 1 = (0 + 1)
12	11	fveq2i 6837	. . . . . 6 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
13	10, 12	eqtri 2763	. . . . 5 ⊢ ℕ = (ℤ_≥‘(0 + 1))
14		id 22	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → 𝑘 = (1 + 𝑚))
15		fveq2 6834	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝐴‘𝑘) = (𝐴‘(1 + 𝑚)))
16	14, 15	oveq12d 7381	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 · (𝐴‘𝑘)) = ((1 + 𝑚) · (𝐴‘(1 + 𝑚))))
17		oveq1 7370	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 − 1) = ((1 + 𝑚) − 1))
18	17	oveq2d 7379	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑦↑(𝑘 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
19	16, 18	oveq12d 7381	. . . . 5 ⊢ (𝑘 = (1 + 𝑚) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) = (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1))))
20		1zzd 12556	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ ℤ)
21		0zd 12534	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 0 ∈ ℤ)
22		nncn 12180	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
23	22	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
24	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐴:ℕ₀⟶ℂ)
25		nnnn0 12442	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
26		ffvelcdm 7029	. . . . . . . 8 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
27	24, 25, 26	syl2an 602	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℂ)
28	23, 27	mulcld 11163	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑘 · (𝐴‘𝑘)) ∈ ℂ)
29		cnvimass 6041	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
30		absf 15298	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
31	30	fdmi 6673	. . . . . . . . . . 11 ⊢ dom abs = ℂ
32	29, 31	sseqtri 3970	. . . . . . . . . 10 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
33	5, 32	eqsstri 3968	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℂ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
35	34	sselda 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
36		nnm1nn0 12476	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
37		expcl 14039	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑘 − 1) ∈ ℕ₀) → (𝑦↑(𝑘 − 1)) ∈ ℂ)
38	35, 36, 37	syl2an 602	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑦↑(𝑘 − 1)) ∈ ℂ)
39	28, 38	mulcld 11163	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) ∈ ℂ)
40	9, 13, 19, 20, 21, 39	isumshft 15802	. . . 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 11094	. . . . . . . 8 ⊢ 1 ∈ ℂ
42		nn0cn 12445	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
43	42	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
44		addcom 11330	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (1 + 𝑚) = (𝑚 + 1))
45	41, 43, 44	sylancr 593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (1 + 𝑚) = (𝑚 + 1))
46	45	fveq2d 6838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝐴‘(1 + 𝑚)) = (𝐴‘(𝑚 + 1)))
47	45, 46	oveq12d 7381	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((1 + 𝑚) · (𝐴‘(1 + 𝑚))) = ((𝑚 + 1) · (𝐴‘(𝑚 + 1))))
48		pncan2 11398	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
49	41, 43, 48	sylancr 593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((1 + 𝑚) − 1) = 𝑚)
50	49	oveq2d 7379	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚))
51	47, 50	oveq12d 7381	. . . . 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 15662	. . . 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 2776	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → Σ𝑚 ∈ ℕ₀ (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚)) = Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))))
54	53	mpteq2dva 5172	. 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 2775	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 396 = wceq 1547 ∈ wcel 2119 {crab 3392 ⊆ wss 3890 ifcif 4461 ↦ cmpt 5160 ^◡ccnv 5624 dom cdm 5625 “ cima 5628 ∘ ccom 5629 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 supcsup 9350 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 ℝ^*cxr 11176 < clt 11177 − cmin 11375 / cdiv 11805 ℕcn 12172 2c2 12234 ℕ₀cn0 12435 ℤ_≥cuz 12786 [,)cico 13298 seqcseq 13961 ↑cexp 14021 abscabs 15194 ⇝ cli 15444 Σcsu 15646 ballcbl 21341 D cdv 25855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-seq 13962 df-exp 14022 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-cmp 23377 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859 df-ulm 26367

This theorem is referenced by: logtayl 26649 binomcxplemdvsum 44806

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