pserdv2 - Metamath Proof Explorer

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Theorem pserdv2 26365

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 26364	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑚 ∈ ℕ₀ (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚))))
9		nn0uz 12771	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
10		nnuz 12772	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
11		1e0p1 12627	. . . . . . 7 ⊢ 1 = (0 + 1)
12	11	fveq2i 6825	. . . . . 6 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
13	10, 12	eqtri 2754	. . . . 5 ⊢ ℕ = (ℤ_≥‘(0 + 1))
14		id 22	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → 𝑘 = (1 + 𝑚))
15		fveq2 6822	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝐴‘𝑘) = (𝐴‘(1 + 𝑚)))
16	14, 15	oveq12d 7364	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 · (𝐴‘𝑘)) = ((1 + 𝑚) · (𝐴‘(1 + 𝑚))))
17		oveq1 7353	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 − 1) = ((1 + 𝑚) − 1))
18	17	oveq2d 7362	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑦↑(𝑘 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
19	16, 18	oveq12d 7364	. . . . 5 ⊢ (𝑘 = (1 + 𝑚) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) = (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1))))
20		1zzd 12500	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ ℤ)
21		0zd 12477	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 0 ∈ ℤ)
22		nncn 12130	. . . . . . . 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 12385	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
26		ffvelcdm 7014	. . . . . . . 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 11129	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑘 · (𝐴‘𝑘)) ∈ ℂ)
29		cnvimass 6031	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
30		absf 15242	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
31	30	fdmi 6662	. . . . . . . . . . 11 ⊢ dom abs = ℂ
32	29, 31	sseqtri 3983	. . . . . . . . . 10 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
33	5, 32	eqsstri 3981	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℂ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
35	34	sselda 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
36		nnm1nn0 12419	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
37		expcl 13983	. . . . . . 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 11129	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) ∈ ℂ)
40	9, 13, 19, 20, 21, 39	isumshft 15743	. . . 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 11061	. . . . . . . 8 ⊢ 1 ∈ ℂ
42		nn0cn 12388	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
43	42	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
44		addcom 11296	. . . . . . . 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 6826	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝐴‘(1 + 𝑚)) = (𝐴‘(𝑚 + 1)))
47	45, 46	oveq12d 7364	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((1 + 𝑚) · (𝐴‘(1 + 𝑚))) = ((𝑚 + 1) · (𝐴‘(𝑚 + 1))))
48		pncan2 11364	. . . . . . . 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 7362	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚))
51	47, 50	oveq12d 7364	. . . . 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 15606	. . . 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 2767	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → Σ𝑚 ∈ ℕ₀ (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚)) = Σ𝑘 ∈ ℕ ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))))
54	53	mpteq2dva 5184	. 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 2766	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 1541 ∈ wcel 2111 {crab 3395 ⊆ wss 3902 ifcif 4475 ↦ cmpt 5172 ^◡ccnv 5615 dom cdm 5616 “ cima 5619 ∘ ccom 5620 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 supcsup 9324 ℂcc 11001 ℝcr 11002 0cc0 11003 1c1 11004 + caddc 11006 · cmul 11008 ℝ^*cxr 11142 < clt 11143 − cmin 11341 / cdiv 11771 ℕcn 12122 2c2 12177 ℕ₀cn0 12378 ℤ_≥cuz 12729 [,)cico 13244 seqcseq 13905 ↑cexp 13965 abscabs 15138 ⇝ cli 15388 Σcsu 15590 ballcbl 21276 D cdv 25789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-seq 13906 df-exp 13966 df-hash 14235 df-shft 14971 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-rlim 15393 df-sum 15591 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-mulg 18978 df-cntz 19227 df-cmn 19692 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-fbas 21286 df-fg 21287 df-cnfld 21290 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-lp 23049 df-perf 23050 df-cn 23140 df-cnp 23141 df-haus 23228 df-cmp 23300 df-tx 23475 df-hmeo 23668 df-fil 23759 df-fm 23851 df-flim 23852 df-flf 23853 df-xms 24233 df-ms 24234 df-tms 24235 df-cncf 24796 df-limc 25792 df-dv 25793 df-ulm 26311

This theorem is referenced by: logtayl 26594 binomcxplemdvsum 44387

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