Theorem fsumabs 15787

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 4004	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumabs.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 4007	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 15675	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	fveq2d 6906	. . . . . . 7 ⊢ (𝑤 = ∅ → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ ∅ 𝐵))
6		sumeq1 15675	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ ∅ (abs‘𝐵))
7	5, 6	breq12d 5165	. . . . . 6 ⊢ (𝑤 = ∅ → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
8	3, 7	imbi12d 343	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵)) ↔ (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵))))
9	8	imbi2d 339	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))))
10		sseq1 4007	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
11		sumeq1 15675	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑥 𝐵)
12	11	fveq2d 6906	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝑥 𝐵))
13		sumeq1 15675	. . . . . . 7 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝑥 (abs‘𝐵))
14	12, 13	breq12d 5165	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)))
15	10, 14	imbi12d 343	. . . . 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 339	. . . 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 4007	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
18		sumeq1 15675	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵)
19	18	fveq2d 6906	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵))
20		sumeq1 15675	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))
21	19, 20	breq12d 5165	. . . . . 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 343	. . . . 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 339	. . . 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 4007	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
25		sumeq1 15675	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
26	25	fveq2d 6906	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝐴 𝐵))
27		sumeq1 15675	. . . . . . 7 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝐴 (abs‘𝐵))
28	26, 27	breq12d 5165	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)))
29	24, 28	imbi12d 343	. . . . 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 339	. . . 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 12351	. . . . . 6 ⊢ 0 ≤ 0
32		sum0 15707	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
33	32	fveq2i 6905	. . . . . . 7 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = (abs‘0)
34		abs0 15272	. . . . . . 7 ⊢ (abs‘0) = 0
35	33, 34	eqtri 2756	. . . . . 6 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = 0
36		sum0 15707	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ (abs‘𝐵) = 0
37	31, 35, 36	3brtr4i 5182	. . . . 5 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)
38	37	2a1i 12	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
39		ssun1 4174	. . . . . . . . . 10 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
40		sstr 3990	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
41	39, 40	mpan 688	. . . . . . . . 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 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝜑)
44	43, 2	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
45		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
46	45	unssad 4189	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
47	44, 46	ssfid 9298	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
48	46	sselda 3982	. . . . . . . . . . . . . . . 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 683	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ ℂ)
51	47, 50	fsumcl 15719	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℂ)
52	51	abscld 15423	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ∈ ℝ)
53	50	abscld 15423	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → (abs‘𝐵) ∈ ℝ)
54	47, 53	fsumrecl 15720	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 (abs‘𝐵) ∈ ℝ)
55	45	unssbd 4190	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
56		vex 3477	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
57	56	snss 4794	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
58	55, 57	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ 𝐴)
59	49	ralrimiva 3143	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
60	43, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
61		nfcsb1v 3919	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵
62	61	nfel1 2916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ
63		csbeq1a 3908	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
64	63	eleq1d 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ))
65	62, 64	rspc 3599	. . . . . . . . . . . . . . 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 15423	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℝ)
68	52, 54, 67	leadd1d 11846	. . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
70		disjsn 4720	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥)
71	69, 70	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
72		eqidd 2729	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
73	44, 45	ssfid 9298	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
74	45	sselda 3982	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝑘 ∈ 𝐴)
75	43, 74, 49	syl2an2r 683	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ ℂ)
76	75	abscld 15423	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℝ)
77	76	recnd 11280	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℂ)
78	71, 72, 73, 77	fsumsplit 15727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)))
79		csbfv2g 6951	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
80	79	elv 3479	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵)
81	67	recnd 11280	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℂ)
82	80, 81	eqeltrid 2833	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌(abs‘𝐵) ∈ ℂ)
83		sumsns 15736	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ V ∧ ⦋𝑦 / 𝑘⦌(abs‘𝐵) ∈ ℂ) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = ⦋𝑦 / 𝑘⦌(abs‘𝐵))
84	56, 82, 83	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = ⦋𝑦 / 𝑘⦌(abs‘𝐵))
85	84, 80	eqtrdi 2784	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
86	85	oveq2d 7442	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
88	87	breq2d 5164	. . . . . . . . . . . 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 15727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵))
91		sumsns 15736	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑦}𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
92	58, 66, 91	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦}𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
93	92	oveq2d 7442	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
94	90, 93	eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
95	94	fveq2d 6906	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) = (abs‘(Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵)))
96	51, 66	abstrid 15443	. . . . . . . . . . . . 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 5174	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
98	73, 75	fsumcl 15719	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 ∈ ℂ)
99	98	abscld 15423	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ)
100	52, 67	readdcld 11281	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ∈ ℝ)
101	73, 76	fsumrecl 15720	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ)
102		letr 11346	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 411	. . . . . . . . 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 412	. . . . . 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 480	. . . 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 9196	. . 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 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  Vcvv 3473  ⦋csb 3894   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326  {csn 4632   class class class wbr 5152  ‘cfv 6553  (class class class)co 7426  Fincfn 8970  ℂcc 11144  ℝcr 11145  0cc0 11146   + caddc 11149   ≤ cle 11287  abscabs 15221  Σcsu 15672

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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224

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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-fzo 13668  df-seq 14007  df-exp 14067  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-sum 15673

This theorem is referenced by:  o1fsum  15799  seqabs  15800  cvgcmpce  15804  mertenslem1  15870  dvfsumabs  25977  mtest  26360  mtestbdd  26361  abelthlem7  26395  fsumharmonic  26964  ftalem1  27025  ftalem5  27029  dchrisumlem2  27443  dchrmusum2  27447  dchrvmasumlem3  27452  dchrvmasumiflem1  27454  dchrisum0lem1  27469  dchrisum0lem2a  27470  mudivsum  27483  mulogsumlem  27484  2vmadivsumlem  27493  selberglem2  27499  selberg3lem1  27510  selberg4lem1  27513  pntrsumbnd  27519  pntrlog2bndlem1  27530  pntrlog2bndlem3  27532  knoppndvlem11  36030  fourierdlem73  45596  etransclem23  45674