fsumsermpt - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumsermpt		Structured version Visualization version GIF version

Theorem fsumsermpt 44281

Description: A finite sum expressed in terms of a partial sum of an infinite series. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
fsumsermpt.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
fsumsermpt.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
fsumsermpt.a	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
fsumsermpt.f	⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴)
fsumsermpt.g	⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))

**Assertion**
Ref	Expression
fsumsermpt	⊢ (𝜑 → 𝐹 = 𝐺)

Distinct variable groups: 𝐴,𝑛 𝑘,𝑀,𝑛 𝑘,𝑍,𝑛 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑛) 𝐴(𝑘) 𝐹(𝑘,𝑛) 𝐺(𝑘,𝑛)

Proof of Theorem fsumsermpt Dummy variables 𝑗 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13934	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑚) ∈ Fin)
2		simpl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝜑)
3		elfzuz 13493	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ_≥‘𝑀))
4		fsumsermpt.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5	3, 4	eleqtrrdi 2844	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍)
6	5	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝑘 ∈ 𝑍)
7		fsumsermpt.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
8	2, 6, 7	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝐴 ∈ ℂ)
9	1, 8	fsumcl 15675	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ)
11	10	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ)
12		fsumsermpt.f	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴)
13		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑀...𝑛) = (𝑀...𝑚))
14	13	sumeq1d 15643	. . . . . 6 ⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (𝑀...𝑛)𝐴 = Σ𝑘 ∈ (𝑀...𝑚)𝐴)
15	14	cbvmptv 5260	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴)
16	12, 15	eqtri 2760	. . . 4 ⊢ 𝐹 = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴)
17	16	fnmpt 6687	. . 3 ⊢ (∀𝑚 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ → 𝐹 Fn 𝑍)
18	11, 17	syl 17	. 2 ⊢ (𝜑 → 𝐹 Fn 𝑍)
19		fsumsermpt.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
21		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
22		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑗
23	22	nfcsb1 3916	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
24	23	nfel1 2919	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ
25	21, 24	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)
26		eleq1w 2816	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
27	26	anbi2d 629	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
28		csbeq1a 3906	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
29	28	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ))
30	27, 29	imbi12d 344	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)))
31	25, 30, 7	chvarfv 2233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)
32		eqid 2732	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴)
33	22, 23, 28, 32	fvmptf 7016	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
34	20, 31, 33	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
35	34, 31	eqeltrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) ∈ ℂ)
36		addcl 11188	. . . . . 6 ⊢ ((𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑗 + 𝑥) ∈ ℂ)
37	36	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑗 + 𝑥) ∈ ℂ)
38	4, 19, 35, 37	seqf 13985	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)):𝑍⟶ℂ)
39	38	ffnd 6715	. . 3 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍)
40		fsumsermpt.g	. . . . 5 ⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))
41	40	a1i 11	. . . 4 ⊢ (𝜑 → 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)))
42	41	fneq1d 6639	. . 3 ⊢ (𝜑 → (𝐺 Fn 𝑍 ↔ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍))
43	39, 42	mpbird 256	. 2 ⊢ (𝜑 → 𝐺 Fn 𝑍)
44		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
45	16	fvmpt2 7006	. . . . 5 ⊢ ((𝑚 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴)
46	44, 10, 45	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴)
47		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑗(𝑀...𝑚)
48		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑘(𝑀...𝑚)
49		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑗𝐴
50	28, 47, 48, 49, 23	cbvsum 15637	. . . . 5 ⊢ Σ𝑘 ∈ (𝑀...𝑚)𝐴 = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴
51	50	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑚)𝐴 = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴)
52	46, 51	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴)
53		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝜑)
54		elfzuz 13493	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ (ℤ_≥‘𝑀))
55	54, 4	eleqtrrdi 2844	. . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ 𝑍)
56	55	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝑗 ∈ 𝑍)
57	53, 56, 34	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
58	57	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑚)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
59		id 22	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ 𝑍)
60	59, 4	eleqtrdi 2843	. . . . 5 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ_≥‘𝑀))
61	60	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ (ℤ_≥‘𝑀))
62	53, 56, 31	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)
63	62	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑚)) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)
64	58, 61, 63	fsumser 15672	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴 = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚))
65	40	fveq1i 6889	. . . . 5 ⊢ (𝐺‘𝑚) = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚)
66	65	eqcomi 2741	. . . 4 ⊢ (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) = (𝐺‘𝑚)
67	66	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) = (𝐺‘𝑚))
68	52, 64, 67	3eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚))
69	18, 43, 68	eqfnfvd 7032	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⦋csb 3892 ↦ cmpt 5230 Fn wfn 6535 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 + caddc 11109 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 seqcseq 13962 Σcsu 15628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629

This theorem is referenced by: ovolval2lem 45345

W3C validator