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 26360

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 26359	. 2 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑦 ∈ 𝑆 ↦ Σ𝑚 ∈ ℕ₀ (((𝑚 + 1) · (𝐴‘(𝑚 + 1))) · (𝑦↑𝑚))))
9		nn0uz 12888	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
10		nnuz 12889	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
11		1e0p1 12743	. . . . . . 7 ⊢ 1 = (0 + 1)
12	11	fveq2i 6894	. . . . . 6 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
13	10, 12	eqtri 2756	. . . . 5 ⊢ ℕ = (ℤ_≥‘(0 + 1))
14		id 22	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → 𝑘 = (1 + 𝑚))
15		fveq2 6891	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝐴‘𝑘) = (𝐴‘(1 + 𝑚)))
16	14, 15	oveq12d 7432	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 · (𝐴‘𝑘)) = ((1 + 𝑚) · (𝐴‘(1 + 𝑚))))
17		oveq1 7421	. . . . . . 7 ⊢ (𝑘 = (1 + 𝑚) → (𝑘 − 1) = ((1 + 𝑚) − 1))
18	17	oveq2d 7430	. . . . . 6 ⊢ (𝑘 = (1 + 𝑚) → (𝑦↑(𝑘 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
19	16, 18	oveq12d 7432	. . . . 5 ⊢ (𝑘 = (1 + 𝑚) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) = (((1 + 𝑚) · (𝐴‘(1 + 𝑚))) · (𝑦↑((1 + 𝑚) − 1))))
20		1zzd 12617	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ ℤ)
21		0zd 12594	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 0 ∈ ℤ)
22		nncn 12244	. . . . . . . 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 12503	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
26		ffvelcdm 7085	. . . . . . . 8 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
27	24, 25, 26	syl2an 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℂ)
28	23, 27	mulcld 11258	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑘 · (𝐴‘𝑘)) ∈ ℂ)
29		cnvimass 6079	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
30		absf 15310	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
31	30	fdmi 6728	. . . . . . . . . . 11 ⊢ dom abs = ℂ
32	29, 31	sseqtri 4014	. . . . . . . . . 10 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
33	5, 32	eqsstri 4012	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℂ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
35	34	sselda 3978	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
36		nnm1nn0 12537	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
37		expcl 14070	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑘 − 1) ∈ ℕ₀) → (𝑦↑(𝑘 − 1)) ∈ ℂ)
38	35, 36, 37	syl2an 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → (𝑦↑(𝑘 − 1)) ∈ ℂ)
39	28, 38	mulcld 11258	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐴‘𝑘)) · (𝑦↑(𝑘 − 1))) ∈ ℂ)
40	9, 13, 19, 20, 21, 39	isumshft 15811	. . . 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 11190	. . . . . . . 8 ⊢ 1 ∈ ℂ
42		nn0cn 12506	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
43	42	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
44		addcom 11424	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (1 + 𝑚) = (𝑚 + 1))
45	41, 43, 44	sylancr 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (1 + 𝑚) = (𝑚 + 1))
46	45	fveq2d 6895	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝐴‘(1 + 𝑚)) = (𝐴‘(𝑚 + 1)))
47	45, 46	oveq12d 7432	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((1 + 𝑚) · (𝐴‘(1 + 𝑚))) = ((𝑚 + 1) · (𝐴‘(𝑚 + 1))))
48		pncan2 11491	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
49	41, 43, 48	sylancr 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((1 + 𝑚) − 1) = 𝑚)
50	49	oveq2d 7430	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑚 ∈ ℕ₀) → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚))
51	47, 50	oveq12d 7432	. . . . 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 15675	. . . 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 2769	. . 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 2768	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 1534 ∈ wcel 2099 {crab 3428 ⊆ wss 3945 ifcif 4524 ↦ cmpt 5225 ^◡ccnv 5671 dom cdm 5672 “ cima 5675 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 supcsup 9457 ℂcc 11130 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 ℝ^*cxr 11271 < clt 11272 − cmin 11468 / cdiv 11895 ℕcn 12236 2c2 12291 ℕ₀cn0 12496 ℤ_≥cuz 12846 [,)cico 13352 seqcseq 13992 ↑cexp 14052 abscabs 15207 ⇝ cli 15454 Σcsu 15658 ballcbl 21259 D cdv 25785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-seq 13993 df-exp 14053 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-lp 23033 df-perf 23034 df-cn 23124 df-cnp 23125 df-haus 23212 df-cmp 23284 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-xms 24219 df-ms 24220 df-tms 24221 df-cncf 24791 df-limc 25788 df-dv 25789 df-ulm 26306

This theorem is referenced by: logtayl 26587 binomcxplemdvsum 43786

W3C validator