Theorem fsumabs 12159

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 3260	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumabs.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3263	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 12048	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	fveq2d 5676	. . . . . . 7 ⊢ (𝑤 = ∅ → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ ∅ 𝐵))
6		sumeq1 12048	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ ∅ (abs‘𝐵))
7	5, 6	breq12d 4124	. . . . . 6 ⊢ (𝑤 = ∅ → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
8	3, 7	imbi12d 234	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵)) ↔ (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵))))
9	8	imbi2d 230	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))))
10		sseq1 3263	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
11		sumeq1 12048	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑥 𝐵)
12	11	fveq2d 5676	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝑥 𝐵))
13		sumeq1 12048	. . . . . . 7 ⊢ (𝑤 = 𝑥 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝑥 (abs‘𝐵))
14	12, 13	breq12d 4124	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)))
15	10, 14	imbi12d 234	. . . . 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 230	. . . 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 3263	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
18		sumeq1 12048	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵)
19	18	fveq2d 5676	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵))
20		sumeq1 12048	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))
21	19, 20	breq12d 4124	. . . . . 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 234	. . . . 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 230	. . . 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 3263	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
25		sumeq1 12048	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
26	25	fveq2d 5676	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (abs‘Σ𝑘 ∈ 𝑤 𝐵) = (abs‘Σ𝑘 ∈ 𝐴 𝐵))
27		sumeq1 12048	. . . . . . 7 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 (abs‘𝐵) = Σ𝑘 ∈ 𝐴 (abs‘𝐵))
28	26, 27	breq12d 4124	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((abs‘Σ𝑘 ∈ 𝑤 𝐵) ≤ Σ𝑘 ∈ 𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)))
29	24, 28	imbi12d 234	. . . . 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 230	. . . 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 9331	. . . . . 6 ⊢ 0 ≤ 0
32		sum0 12082	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
33	32	fveq2i 5675	. . . . . . 7 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = (abs‘0)
34		abs0 11751	. . . . . . 7 ⊢ (abs‘0) = 0
35	33, 34	eqtri 2255	. . . . . 6 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) = 0
36		sum0 12082	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ (abs‘𝐵) = 0
37	31, 35, 36	3brtr4i 4141	. . . . 5 ⊢ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)
38	37	2a1i 27	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
39		ssun1 3384	. . . . . . . . 9 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
40		sstr 3248	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
41	39, 40	mpan 424	. . . . . . . 8 ⊢ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
42	41	imim1i 60	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵)))
43		simplrl 537	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
44		simpll 527	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝜑)
45		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
46	45	unssad 3398	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
47	46	sselda 3240	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐴)
48		fsumabs.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
49	44, 47, 48	syl2an2r 599	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ ℂ)
50	43, 49	fsumcl 12094	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℂ)
51	50	abscld 11874	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ 𝑥 𝐵) ∈ ℝ)
52	49	abscld 11874	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ 𝑥) → (abs‘𝐵) ∈ ℝ)
53	43, 52	fsumrecl 12095	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ 𝑥 (abs‘𝐵) ∈ ℝ)
54		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
55	54	unssbd 3399	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
56		vex 2818	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
57	56	snss 3831	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
58	55, 57	sylibr 134	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ 𝐴)
59	58	adantlrl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ 𝐴)
60	48	ralrimiva 2617	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
61	60	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
62		nfcsb1v 3173	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵
63	62	nfel1 2397	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ
64		csbeq1a 3149	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
65	64	eleq1d 2303	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ))
66	63, 65	rspc 2917	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ))
67	59, 61, 66	sylc 62	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ)
68	67	abscld 11874	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℝ)
69	51, 53, 68	leadd1d 8818	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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))))
70		simplr 529	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
71	70	adantlrl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
72		disjsn 3753	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥)
73	71, 72	sylibr 134	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
74		eqidd 2235	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
75	56	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ V)
76		unsnfi 7181	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑥) → (𝑥 ∪ {𝑦}) ∈ Fin)
77	43, 75, 71, 76	syl3anc 1274	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
78	45	sselda 3240	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝑘 ∈ 𝐴)
79	44, 78, 48	syl2an2r 599	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ ℂ)
80	79	abscld 11874	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℝ)
81	80	recnd 8307	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℂ)
82	73, 74, 77, 81	fsumsplit 12101	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)))
83		csbfv2g 5713	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
84	83	elv 2819	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋𝑦 / 𝑘⦌(abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵)
85	60	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
86	58, 85, 66	sylc 62	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ)
87	86	abscld 11874	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℝ)
88	87	recnd 8307	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘⦋𝑦 / 𝑘⦌𝐵) ∈ ℂ)
89	84, 88	eqeltrid 2321	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌(abs‘𝐵) ∈ ℂ)
90		sumsns 12109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ V ∧ ⦋𝑦 / 𝑘⦌(abs‘𝐵) ∈ ℂ) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = ⦋𝑦 / 𝑘⦌(abs‘𝐵))
91	56, 89, 90	sylancr 414	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = ⦋𝑦 / 𝑘⦌(abs‘𝐵))
92	91, 84	eqtrdi 2283	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = (abs‘⦋𝑦 / 𝑘⦌𝐵))
93	92	oveq2d 6068	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
94	93	adantlrl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
95	82, 94	eqtrd 2267	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘 ∈ 𝑥 (abs‘𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
96	95	breq2d 4123	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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))))
97	69, 96	bitr4d 191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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)))
98	70, 72	sylibr 134	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
99	98	adantlrl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
100	99, 74, 77, 79	fsumsplit 12101	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵))
101		sumsns 12109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑦}𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
102	58, 86, 101	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦}𝐵 = ⦋𝑦 / 𝑘⦌𝐵)
103	102	oveq2d 6068	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
104	103	adantlrl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘 ∈ 𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
105	100, 104	eqtrd 2267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵))
106	105	fveq2d 5676	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) = (abs‘(Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵)))
107	86	adantlrl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ⦋𝑦 / 𝑘⦌𝐵 ∈ ℂ)
108	50, 107	abstrid 11889	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘(Σ𝑘 ∈ 𝑥 𝐵 + ⦋𝑦 / 𝑘⦌𝐵)) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
109	106, 108	eqbrtrd 4133	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)))
110	77, 79	fsumcl 12094	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 ∈ ℂ)
111	110	abscld 11874	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ)
112	51, 68	readdcld 8308	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) + (abs‘⦋𝑦 / 𝑘⦌𝐵)) ∈ ℝ)
113	77, 80	fsumrecl 12095	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ)
114		letr 8361	. . . . . . . . . . . 12 ⊢ (((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)))
115	111, 112, 113, 114	syl3anc 1274	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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)))
116	109, 115	mpand 429	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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)))
117	97, 116	sylbid 150	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
118	117	ex 115	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ((abs‘Σ𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
119	118	a2d 26	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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))))
120	42, 119	syl5 32	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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))))
121	120	expcom 116	. . . . 5 ⊢ ((𝑥 ∈ Fin ∧ ¬ 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)))))
122	121	a2d 26	. . . 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)))))
123	9, 16, 23, 30, 38, 122	findcard2s 7149	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵))))
124	2, 123	mpcom 36	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵)))
125	1, 124	mpi 15	1 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815  ⦋csb 3140   ∪ cun 3211   ∩ cin 3212   ⊆ wss 3213  ∅c0 3510  {csn 3691   class class class wbr 4111  ‘cfv 5354  (class class class)co 6052  Fincfn 6977  ℂcc 8130  ℝcr 8131  0cc0 8132   + caddc 8135   ≤ cle 8314  abscabs 11690  Σcsu 12046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fzo 10484  df-seqfrec 10817  df-exp 10908  df-ihash 11147  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-clim 11972  df-sumdc 12047

This theorem is referenced by:  iserabs  12169  cvgratnnlemabsle  12221  mertenslemi1  12229