fsumsermpt - Mathbox for Glauco Siliprandi

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Theorem fsumsermpt 45029

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 13968	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑚) ∈ Fin)
2		simpl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝜑)
3		elfzuz 13527	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ_≥‘𝑀))
4		fsumsermpt.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5	3, 4	eleqtrrdi 2836	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍)
6	5	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝑘 ∈ 𝑍)
7		fsumsermpt.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
8	2, 6, 7	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝐴 ∈ ℂ)
9	1, 8	fsumcl 15709	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ)
10	9	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ)
11	10	ralrimiva 3136	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ)
12		fsumsermpt.f	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴)
13		oveq2 7423	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑀...𝑛) = (𝑀...𝑚))
14	13	sumeq1d 15677	. . . . . 6 ⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (𝑀...𝑛)𝐴 = Σ𝑘 ∈ (𝑀...𝑚)𝐴)
15	14	cbvmptv 5256	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴)
16	12, 15	eqtri 2753	. . . 4 ⊢ 𝐹 = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴)
17	16	fnmpt 6689	. . 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 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
21		nfv 1909	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
22		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑗
23	22	nfcsb1 3909	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
24	23	nfel1 2909	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ
25	21, 24	nfim 1891	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)
26		eleq1w 2808	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
27	26	anbi2d 628	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
28		csbeq1a 3899	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
29	28	eleq1d 2810	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ))
30	27, 29	imbi12d 343	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)))
31	25, 30, 7	chvarfv 2228	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)
32		eqid 2725	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴)
33	22, 23, 28, 32	fvmptf 7020	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
34	20, 31, 33	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
35	34, 31	eqeltrd 2825	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) ∈ ℂ)
36		addcl 11218	. . . . . 6 ⊢ ((𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑗 + 𝑥) ∈ ℂ)
37	36	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑗 + 𝑥) ∈ ℂ)
38	4, 19, 35, 37	seqf 14018	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)):𝑍⟶ℂ)
39	38	ffnd 6717	. . 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 6641	. . 3 ⊢ (𝜑 → (𝐺 Fn 𝑍 ↔ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍))
43	39, 42	mpbird 256	. 2 ⊢ (𝜑 → 𝐺 Fn 𝑍)
44		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
45	16	fvmpt2 7010	. . . . 5 ⊢ ((𝑚 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴)
46	44, 10, 45	syl2anc 582	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴)
47		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑗(𝑀...𝑚)
48		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑘(𝑀...𝑚)
49		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑗𝐴
50	28, 47, 48, 49, 23	cbvsum 15671	. . . . 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 2765	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴)
53		simpl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝜑)
54		elfzuz 13527	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ (ℤ_≥‘𝑀))
55	54, 4	eleqtrrdi 2836	. . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ 𝑍)
56	55	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝑗 ∈ 𝑍)
57	53, 56, 34	syl2anc 582	. . . . 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 2835	. . . . 5 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ_≥‘𝑀))
61	60	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ (ℤ_≥‘𝑀))
62	53, 56, 31	syl2anc 582	. . . . 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 15706	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴 = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚))
65	40	fveq1i 6892	. . . . 5 ⊢ (𝐺‘𝑚) = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚)
66	65	eqcomi 2734	. . . 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 2769	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚))
69	18, 43, 68	eqfnfvd 7037	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ⦋csb 3885 ↦ cmpt 5226 Fn wfn 6537 ‘cfv 6542 (class class class)co 7415 ℂcc 11134 + caddc 11139 ℤcz 12586 ℤ_≥cuz 12850 ...cfz 13514 seqcseq 13996 Σcsu 15662

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663

This theorem is referenced by: ovolval2lem 46093

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