Theorem fsumabs 14993

Description: Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)

Hypotheses

Ref	Expression
fsumabs.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumabs.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fsumabs	⊢ (𝜑 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵))

Distinct variable groups:   𝐴,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fsumabs

Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3916	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumabs.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3919	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 14883	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	fveq2d 6549	. . . . . . 7 ⊢ (𝑤 = ∅ → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ ∅ 𝐵))
6		sumeq1 14883	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ ∅ (abs‘𝐵))
7	5, 6	breq12d 4981	. . . . . 6 ⊢ (𝑤 = ∅ → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
8	3, 7	imbi12d 346	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵)) ↔ (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵))))
9	8	imbi2d 342	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))))
10		sseq1 3919	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
11		sumeq1 14883	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑥 𝐵)
12	11	fveq2d 6549	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝑥 𝐵))
13		sumeq1 14883	. . . . . . 7 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝑥 (abs‘𝐵))
14	12, 13	breq12d 4981	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)))
15	10, 14	imbi12d 346	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵)) ↔ (𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵))))
16	15	imbi2d 342	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝜑 → (𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)))))
17		sseq1 3919	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
18		sumeq1 14883	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵)
19	18	fveq2d 6549	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵))
20		sumeq1 14883	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))
21	19, 20	breq12d 4981	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
22	17, 21	imbi12d 346	. . . . 5 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ((𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵)) ↔ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
23	22	imbi2d 342	. . . 4 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ((𝜑 → (𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵))) ↔ (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
24		sseq1 3919	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
25		sumeq1 14883	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
26	25	fveq2d 6549	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝐴 𝐵))
27		sumeq1 14883	. . . . . . 7 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝐴 (abs‘𝐵))
28	26, 27	breq12d 4981	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)))
29	24, 28	imbi12d 346	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵)) ↔ (𝐴 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵))))
30	29	imbi2d 342	. . . 4 ⊢ (𝑤 = 𝐴 → ((𝜑 → (𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)))))
31		0le0 11592	. . . . . 6 ⊢ 0 ≤ 0
32		sum0 14915	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
33	32	fveq2i 6548	. . . . . . 7 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = (abs‘0)
34		abs0 14483	. . . . . . 7 ⊢ (abs‘0) = 0
35	33, 34	eqtri 2821	. . . . . 6 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = 0
36		sum0 14915	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ (abs‘𝐵) = 0
37	31, 35, 36	3brtr4i 4998	. . . . 5 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)
38	37	2a1i 12	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
39		ssun1 4075	. . . . . . . . . 10 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
40		sstr 3903	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
41	39, 40	mpan 686	. . . . . . . . 9 ⊢ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
42	41	imim1i 63	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)))
43		simpll 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝜑)
44	43, 2	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
45		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
46	45	unssad 4090	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
47	44, 46	ssfid 8594	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
48	46	sselda 3895	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐴)
49		fsumabs.2	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
50	43, 48, 49	syl2an2r 681	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ ℂ)
51	47, 50	fsumcl 14927	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℂ)
52	51	abscld 14634	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ∈ ℝ)
53	50	abscld 14634	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → (abs‘𝐵) ∈ ℝ)
54	47, 53	fsumrecl 14928	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 (abs‘𝐵) ∈ ℝ)
55	45	unssbd 4091	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
56		vex 3443	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
57	56	snss 4631	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
58	55, 57	sylibr 235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ 𝐴)
59	49	ralrimiva 3151	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
60	43, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
61		nfcsb1v 3839	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵
62	61	nfel1 2965	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ
63		csbeq1a 3830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
64	63	eleq1d 2869	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ))
65	62, 64	rspc 3555	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ))
66	58, 60, 65	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ)
67	66	abscld 14634	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℝ)
68	52, 54, 67	leadd1d 11088	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵) ↔ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ≤ (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵))))
69		simplr 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
70		disjsn 4560	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥)
71	69, 70	sylibr 235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
72		eqidd 2798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
73	44, 45	ssfid 8594	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
74	45	sselda 3895	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝑘 ∈ 𝐴)
75	43, 74, 49	syl2an2r 681	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ ℂ)
76	75	abscld 14634	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℝ)
77	76	recnd 10522	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℂ)
78	71, 72, 73, 77	fsumsplit 14934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)))
79		csbfv2g 6589	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
80	79	elv 3445	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵)
81	67	recnd 10522	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℂ)
82	80, 81	syl5eqel 2889	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌(abs‘𝐵) ∈ ℂ)
83		sumsns 14942	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ V ∧ ⦋𝑦 / 𝑘⦌(abs‘𝐵) ∈ ℂ) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = ⦋𝑦 / 𝑘⦌(abs‘𝐵))
84	56, 82, 83	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = ⦋𝑦 / 𝑘⦌(abs‘𝐵))
85	84, 80	syl6eq 2849	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
86	85	oveq2d 7039	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
87	78, 86	eqtrd 2833	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
88	87	breq2d 4980	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ↔ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ≤ (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵))))
89	68, 88	bitr4d 283	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵) ↔ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
90	71, 72, 73, 75	fsumsplit 14934	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵))
91		sumsns 14942	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑦}𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
92	58, 66, 91	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦}𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
93	92	oveq2d 7039	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
94	90, 93	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
95	94	fveq2d 6549	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) = (abs‘(Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵)))
96	51, 66	abstrid 14654	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘(Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵)) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
97	95, 96	eqbrtrd 4990	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
98	73, 75	fsumcl 14927	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 ∈ ℂ)
99	98	abscld 14634	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ)
100	52, 67	readdcld 10523	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ∈ ℝ)
101	73, 76	fsumrecl 14928	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ)
102		letr 10587	. . . . . . . . . . . . 13 ⊢ (((abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ ∧ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ∈ ℝ ∧ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ) → (((abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ∧ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
103	99, 100, 101, 102	syl3anc 1364	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (((abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ∧ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
104	97, 103	mpand 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
105	89, 104	sylbid 241	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
106	105	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
107	106	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) → (((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
108	42, 107	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) → ((𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
109	108	expcom 414	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑥 → (𝜑 → ((𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
110	109	a2d 29	. . . . 5 ⊢ (¬ 𝑦 ∈ 𝑥 → ((𝜑 → (𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵))) → (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
111	110	adantl 482	. . . 4 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥) → ((𝜑 → (𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵))) → (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
112	9, 16, 23, 30, 38, 111	findcard2s 8612	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵))))
113	2, 112	mpcom 38	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)))
114	1, 113	mpi 20	1 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1525   ∈ wcel 2083  ∀wral 3107  Vcvv 3440  ⦋csb 3817   ∪ cun 3863   ∩ cin 3864   ⊆ wss 3865  ∅c0 4217  {csn 4478   class class class wbr 4968  ‘cfv 6232  (class class class)co 7023  Fincfn 8364  ℂcc 10388  ℝcr 10389  0cc0 10390   + caddc 10393   ≤ cle 10529  abscabs 14431  Σcsu 14880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-oi 8827  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-fz 12747  df-fzo 12888  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-sum 14881

This theorem is referenced by:  o1fsum  15005  seqabs  15006  cvgcmpce  15010  mertenslem1  15077  dvfsumabs  24307  mtest  24679  mtestbdd  24680  abelthlem7  24713  fsumharmonic  25275  ftalem1  25336  ftalem5  25340  dchrisumlem2  25752  dchrmusum2  25756  dchrvmasumlem3  25761  dchrvmasumiflem1  25763  dchrisum0lem1  25778  dchrisum0lem2a  25779  mudivsum  25792  mulogsumlem  25793  2vmadivsumlem  25802  selberglem2  25808  selberg3lem1  25819  selberg4lem1  25822  pntrsumbnd  25828  pntrlog2bndlem1  25839  pntrlog2bndlem3  25841  knoppndvlem11  33472  fourierdlem73  42028  etransclem23  42106