Theorem fsumabs 15513

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 3943	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumabs.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3946	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 15400	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	fveq2d 6778	. . . . . . 7 ⊢ (𝑤 = ∅ → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ ∅ 𝐵))
6		sumeq1 15400	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ ∅ (abs‘𝐵))
7	5, 6	breq12d 5087	. . . . . 6 ⊢ (𝑤 = ∅ → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
8	3, 7	imbi12d 345	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵)) ↔ (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵))))
9	8	imbi2d 341	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))))
10		sseq1 3946	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
11		sumeq1 15400	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑥 𝐵)
12	11	fveq2d 6778	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝑥 𝐵))
13		sumeq1 15400	. . . . . . 7 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝑥 (abs‘𝐵))
14	12, 13	breq12d 5087	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)))
15	10, 14	imbi12d 345	. . . . 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 341	. . . 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 3946	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
18		sumeq1 15400	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵)
19	18	fveq2d 6778	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵))
20		sumeq1 15400	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))
21	19, 20	breq12d 5087	. . . . . 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 345	. . . . 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 341	. . . 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 3946	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
25		sumeq1 15400	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
26	25	fveq2d 6778	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝐴 𝐵))
27		sumeq1 15400	. . . . . . 7 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝐴 (abs‘𝐵))
28	26, 27	breq12d 5087	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)))
29	24, 28	imbi12d 345	. . . . 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 341	. . . 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 12074	. . . . . 6 ⊢ 0 ≤ 0
32		sum0 15433	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
33	32	fveq2i 6777	. . . . . . 7 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = (abs‘0)
34		abs0 14997	. . . . . . 7 ⊢ (abs‘0) = 0
35	33, 34	eqtri 2766	. . . . . 6 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = 0
36		sum0 15433	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ (abs‘𝐵) = 0
37	31, 35, 36	3brtr4i 5104	. . . . 5 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)
38	37	2a1i 12	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
39		ssun1 4106	. . . . . . . . . 10 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
40		sstr 3929	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
41	39, 40	mpan 687	. . . . . . . . 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 764	. . . . . . . . . . . . . . . . 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 4121	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
47	44, 46	ssfid 9042	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
48	46	sselda 3921	. . . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ ℂ)
51	47, 50	fsumcl 15445	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℂ)
52	51	abscld 15148	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ∈ ℝ)
53	50	abscld 15148	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → (abs‘𝐵) ∈ ℝ)
54	47, 53	fsumrecl 15446	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 (abs‘𝐵) ∈ ℝ)
55	45	unssbd 4122	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
56		vex 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
57	56	snss 4719	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
58	55, 57	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ 𝐴)
59	49	ralrimiva 3103	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
60	43, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
61		nfcsb1v 3857	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵
62	61	nfel1 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ
63		csbeq1a 3846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
64	63	eleq1d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ))
65	62, 64	rspc 3549	. . . . . . . . . . . . . . 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 15148	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℝ)
68	52, 54, 67	leadd1d 11569	. . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
70		disjsn 4647	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥)
71	69, 70	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
72		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
73	44, 45	ssfid 9042	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
74	45	sselda 3921	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝑘 ∈ 𝐴)
75	43, 74, 49	syl2an2r 682	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ ℂ)
76	75	abscld 15148	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℝ)
77	76	recnd 11003	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℂ)
78	71, 72, 73, 77	fsumsplit 15453	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)))
79		csbfv2g 6818	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
80	79	elv 3438	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵)
81	67	recnd 11003	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℂ)
82	80, 81	eqeltrid 2843	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌(abs‘𝐵) ∈ ℂ)
83		sumsns 15462	. . . . . . . . . . . . . . . . 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	eqtrdi 2794	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
86	85	oveq2d 7291	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
88	87	breq2d 5086	. . . . . . . . . . . 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 281	. . . . . . . . . . 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 15453	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵))
91		sumsns 15462	. . . . . . . . . . . . . . . . 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 7291	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
94	90, 93	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
95	94	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) = (abs‘(Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵)))
96	51, 66	abstrid 15168	. . . . . . . . . . . . 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 5096	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
98	73, 75	fsumcl 15445	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 ∈ ℂ)
99	98	abscld 15148	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ)
100	52, 67	readdcld 11004	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ∈ ℝ)
101	73, 76	fsumrecl 15446	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ)
102		letr 11069	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 692	. . . . . . . . . . 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 239	. . . . . . . . . 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 8948	. . 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 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ⦋csb 3832   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ℂcc 10869  ℝcr 10870  0cc0 10871   + caddc 10874   ≤ cle 11010  abscabs 14945  Σcsu 15397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398

This theorem is referenced by:  o1fsum  15525  seqabs  15526  cvgcmpce  15530  mertenslem1  15596  dvfsumabs  25187  mtest  25563  mtestbdd  25564  abelthlem7  25597  fsumharmonic  26161  ftalem1  26222  ftalem5  26226  dchrisumlem2  26638  dchrmusum2  26642  dchrvmasumlem3  26647  dchrvmasumiflem1  26649  dchrisum0lem1  26664  dchrisum0lem2a  26665  mudivsum  26678  mulogsumlem  26679  2vmadivsumlem  26688  selberglem2  26694  selberg3lem1  26705  selberg4lem1  26708  pntrsumbnd  26714  pntrlog2bndlem1  26725  pntrlog2bndlem3  26727  knoppndvlem11  34702  fourierdlem73  43720  etransclem23  43798