pserdv2 - Metamath Proof Explorer

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Theorem pserdv2 26474

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 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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