Theorem fsumabs 6981

Description: Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values.

Assertion

Ref	Expression
fsumabs	⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...N)A ∈ ℂ) → (abs ‘Σk ∈ (M...N)A) ≤ Σk ∈ (M...N)(abs ‘A))

Distinct variable groups:   k,M   k,N

Proof of Theorem fsumabs

Step	Hyp	Ref	Expression
1		opreq2 3954	. . . . 5 ⊢ (j = M → (M...j) = (M...M))
2	1	raleq1d 1781	. . . 4 ⊢ (j = M → (∀k ∈ (M...j)A ∈ ℂ ↔ ∀k ∈ (M...M)A ∈ ℂ))
3	1	sumeq1d 6928	. . . . . 6 ⊢ (j = M → Σk ∈ (M...j)A = Σk ∈ (M...M)A)
4	3	fveq2d 3713	. . . . 5 ⊢ (j = M → (abs ‘Σk ∈ (M...j)A) = (abs ‘Σk ∈ (M...M)A))
5	1	sumeq1d 6928	. . . . 5 ⊢ (j = M → Σk ∈ (M...j)(abs ‘A) = Σk ∈ (M...M)(abs ‘A))
6	4, 5	breq12d 2621	. . . 4 ⊢ (j = M → ((abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A) ↔ (abs ‘Σk ∈ (M...M)A) ≤ Σk ∈ (M...M)(abs ‘A)))
7	2, 6	imbi12d 624	. . 3 ⊢ (j = M → ((∀k ∈ (M...j)A ∈ ℂ → (abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A)) ↔ (∀k ∈ (M...M)A ∈ ℂ → (abs ‘Σk ∈ (M...M)A) ≤ Σk ∈ (M...M)(abs ‘A))))
8		opreq2 3954	. . . . 5 ⊢ (j = m → (M...j) = (M...m))
9	8	raleq1d 1781	. . . 4 ⊢ (j = m → (∀k ∈ (M...j)A ∈ ℂ ↔ ∀k ∈ (M...m)A ∈ ℂ))
10	8	sumeq1d 6928	. . . . . 6 ⊢ (j = m → Σk ∈ (M...j)A = Σk ∈ (M...m)A)
11	10	fveq2d 3713	. . . . 5 ⊢ (j = m → (abs ‘Σk ∈ (M...j)A) = (abs ‘Σk ∈ (M...m)A))
12	8	sumeq1d 6928	. . . . 5 ⊢ (j = m → Σk ∈ (M...j)(abs ‘A) = Σk ∈ (M...m)(abs ‘A))
13	11, 12	breq12d 2621	. . . 4 ⊢ (j = m → ((abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A) ↔ (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)))
14	9, 13	imbi12d 624	. . 3 ⊢ (j = m → ((∀k ∈ (M...j)A ∈ ℂ → (abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A)) ↔ (∀k ∈ (M...m)A ∈ ℂ → (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A))))
15		opreq2 3954	. . . . 5 ⊢ (j = (m + 1) → (M...j) = (M...(m + 1)))
16	15	raleq1d 1781	. . . 4 ⊢ (j = (m + 1) → (∀k ∈ (M...j)A ∈ ℂ ↔ ∀k ∈ (M...(m + 1))A ∈ ℂ))
17	15	sumeq1d 6928	. . . . . 6 ⊢ (j = (m + 1) → Σk ∈ (M...j)A = Σk ∈ (M...(m + 1))A)
18	17	fveq2d 3713	. . . . 5 ⊢ (j = (m + 1) → (abs ‘Σk ∈ (M...j)A) = (abs ‘Σk ∈ (M...(m + 1))A))
19	15	sumeq1d 6928	. . . . 5 ⊢ (j = (m + 1) → Σk ∈ (M...j)(abs ‘A) = Σk ∈ (M...(m + 1))(abs ‘A))
20	18, 19	breq12d 2621	. . . 4 ⊢ (j = (m + 1) → ((abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A) ↔ (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A)))
21	16, 20	imbi12d 624	. . 3 ⊢ (j = (m + 1) → ((∀k ∈ (M...j)A ∈ ℂ → (abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A)) ↔ (∀k ∈ (M...(m + 1))A ∈ ℂ → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A))))
22		opreq2 3954	. . . . 5 ⊢ (j = N → (M...j) = (M...N))
23	22	raleq1d 1781	. . . 4 ⊢ (j = N → (∀k ∈ (M...j)A ∈ ℂ ↔ ∀k ∈ (M...N)A ∈ ℂ))
24	22	sumeq1d 6928	. . . . . 6 ⊢ (j = N → Σk ∈ (M...j)A = Σk ∈ (M...N)A)
25	24	fveq2d 3713	. . . . 5 ⊢ (j = N → (abs ‘Σk ∈ (M...j)A) = (abs ‘Σk ∈ (M...N)A))
26	22	sumeq1d 6928	. . . . 5 ⊢ (j = N → Σk ∈ (M...j)(abs ‘A) = Σk ∈ (M...N)(abs ‘A))
27	25, 26	breq12d 2621	. . . 4 ⊢ (j = N → ((abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A) ↔ (abs ‘Σk ∈ (M...N)A) ≤ Σk ∈ (M...N)(abs ‘A)))
28	23, 27	imbi12d 624	. . 3 ⊢ (j = N → ((∀k ∈ (M...j)A ∈ ℂ → (abs ‘Σk ∈ (M...j)A) ≤ Σk ∈ (M...j)(abs ‘A)) ↔ (∀k ∈ (M...N)A ∈ ℂ → (abs ‘Σk ∈ (M...N)A) ≤ Σk ∈ (M...N)(abs ‘A))))
29		csbfv2g 3728	. . . . . . 7 ⊢ (M ∈ ℤ → [M / k](abs ‘A) = (abs ‘[M / k]A))
30	29	adantr 389	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → [M / k](abs ‘A) = (abs ‘[M / k]A))
31		ra4csbela 2032	. . . . . . . 8 ⊢ ((M ∈ (M...M) ⋀ ∀k ∈ (M...M)(abs ‘A) ∈ ℝ) → [M / k](abs ‘A) ∈ ℝ)
32		elfz3t 6423	. . . . . . . 8 ⊢ (M ∈ ℤ → M ∈ (M...M))
33		absclt 6768	. . . . . . . . 9 ⊢ (A ∈ ℂ → (abs ‘A) ∈ ℝ)
34	33	r19.20si 1698	. . . . . . . 8 ⊢ (∀k ∈ (M...M)A ∈ ℂ → ∀k ∈ (M...M)(abs ‘A) ∈ ℝ)
35	31, 32, 34	syl2an 454	. . . . . . 7 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → [M / k](abs ‘A) ∈ ℝ)
36		leidt 5504	. . . . . . 7 ⊢ ([M / k](abs ‘A) ∈ ℝ → [M / k](abs ‘A) ≤ [M / k](abs ‘A))
37	35, 36	syl 10	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → [M / k](abs ‘A) ≤ [M / k](abs ‘A))
38	30, 37	eqbrtrrd 2627	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → (abs ‘[M / k]A) ≤ [M / k](abs ‘A))
39		fsum1s 6947	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → Σk ∈ (M...M)A = [M / k]A)
40	39	fveq2d 3713	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → (abs ‘Σk ∈ (M...M)A) = (abs ‘[M / k]A))
41		fsum1s 6947	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(abs ‘A) ∈ ℂ) → Σk ∈ (M...M)(abs ‘A) = [M / k](abs ‘A))
42	33	recnd 5287	. . . . . . 7 ⊢ (A ∈ ℂ → (abs ‘A) ∈ ℂ)
43	42	r19.20si 1698	. . . . . 6 ⊢ (∀k ∈ (M...M)A ∈ ℂ → ∀k ∈ (M...M)(abs ‘A) ∈ ℂ)
44	41, 43	sylan2 451	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → Σk ∈ (M...M)(abs ‘A) = [M / k](abs ‘A))
45	38, 40, 44	3brtr4d 2635	. . . 4 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → (abs ‘Σk ∈ (M...M)A) ≤ Σk ∈ (M...M)(abs ‘A))
46	45	ex 373	. . 3 ⊢ (M ∈ ℤ → (∀k ∈ (M...M)A ∈ ℂ → (abs ‘Σk ∈ (M...M)A) ≤ Σk ∈ (M...M)(abs ‘A)))
47		fzssp1t 6438	. . . . . . . . 9 ⊢ ((M ∈ ℤ ⋀ m ∈ ℤ) → (M...m) ⊆ (M...(m + 1)))
48		eluzel2 6356	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → M ∈ ℤ)
49		eluzelz 6355	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → m ∈ ℤ)
50	47, 48, 49	sylanc 471	. . . . . . . 8 ⊢ (m ∈ (ℤ≥ ‘M) → (M...m) ⊆ (M...(m + 1)))
51	50	sseld 2057	. . . . . . 7 ⊢ (m ∈ (ℤ≥ ‘M) → (k ∈ (M...m) → k ∈ (M...(m + 1))))
52	51	imim1d 28	. . . . . 6 ⊢ (m ∈ (ℤ≥ ‘M) → ((k ∈ (M...(m + 1)) → A ∈ ℂ) → (k ∈ (M...m) → A ∈ ℂ)))
53	52	r19.20dv2 1703	. . . . 5 ⊢ (m ∈ (ℤ≥ ‘M) → (∀k ∈ (M...(m + 1))A ∈ ℂ → ∀k ∈ (M...m)A ∈ ℂ))
54	53	imim1d 28	. . . 4 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...m)A ∈ ℂ → (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → (∀k ∈ (M...(m + 1))A ∈ ℂ → (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A))))
55		abstrit 6835	. . . . . . . . . 10 ⊢ ((Σk ∈ (M...m)A ∈ ℂ ⋀ [(m + 1) / k]A ∈ ℂ) → (abs ‘(Σk ∈ (M...m)A + [(m + 1) / k]A)) ≤ ((abs ‘Σk ∈ (M...m)A) + (abs ‘[(m + 1) / k]A)))
56	53	imp 350	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → ∀k ∈ (M...m)A ∈ ℂ)
57		fsumclt 6953	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)A ∈ ℂ) → Σk ∈ (M...m)A ∈ ℂ)
58	56, 57	syldan 467	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...m)A ∈ ℂ)
59		ra4csbela 2032	. . . . . . . . . . 11 ⊢ (((m + 1) ∈ (M...(m + 1)) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → [(m + 1) / k]A ∈ ℂ)
60		peano2uz 6379	. . . . . . . . . . . 12 ⊢ (m ∈ (ℤ≥ ‘M) → (m + 1) ∈ (ℤ≥ ‘M))
61		eluzfz2t 6421	. . . . . . . . . . . 12 ⊢ ((m + 1) ∈ (ℤ≥ ‘M) → (m + 1) ∈ (M...(m + 1)))
62	60, 61	syl 10	. . . . . . . . . . 11 ⊢ (m ∈ (ℤ≥ ‘M) → (m + 1) ∈ (M...(m + 1)))
63	59, 62	sylan 448	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → [(m + 1) / k]A ∈ ℂ)
64	55, 58, 63	sylanc 471	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → (abs ‘(Σk ∈ (M...m)A + [(m + 1) / k]A)) ≤ ((abs ‘Σk ∈ (M...m)A) + (abs ‘[(m + 1) / k]A)))
65		fsump1s 6951	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...(m + 1))A = (Σk ∈ (M...m)A + [(m + 1) / k]A))
66	65	fveq2d 3713	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → (abs ‘Σk ∈ (M...(m + 1))A) = (abs ‘(Σk ∈ (M...m)A + [(m + 1) / k]A)))
67		oprex 3968	. . . . . . . . . . . 12 ⊢ (m + 1) ∈ V
68		csbfv2g 3728	. . . . . . . . . . . 12 ⊢ ((m + 1) ∈ V → [(m + 1) / k](abs ‘A) = (abs ‘[(m + 1) / k]A))
69	67, 68	ax-mp 7	. . . . . . . . . . 11 ⊢ [(m + 1) / k](abs ‘A) = (abs ‘[(m + 1) / k]A)
70	69	a1i 8	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → [(m + 1) / k](abs ‘A) = (abs ‘[(m + 1) / k]A))
71	70	opreq2d 3961	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) = ((abs ‘Σk ∈ (M...m)A) + (abs ‘[(m + 1) / k]A)))
72	64, 66, 71	3brtr4d 2635	. . . . . . . 8 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → (abs ‘Σk ∈ (M...(m + 1))A) ≤ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)))
73	72	adantr 389	. . . . . . 7 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) ⋀ (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → (abs ‘Σk ∈ (M...(m + 1))A) ≤ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)))
74		leadd1t 5599	. . . . . . . . . 10 ⊢ (((abs ‘Σk ∈ (M...m)A) ∈ ℝ ⋀ Σk ∈ (M...m)(abs ‘A) ∈ ℝ ⋀ [(m + 1) / k](abs ‘A) ∈ ℝ) → ((abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A) ↔ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ≤ (Σk ∈ (M...m)(abs ‘A) + [(m + 1) / k](abs ‘A))))
75		absclt 6768	. . . . . . . . . . 11 ⊢ (Σk ∈ (M...m)A ∈ ℂ → (abs ‘Σk ∈ (M...m)A) ∈ ℝ)
76	58, 75	syl 10	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → (abs ‘Σk ∈ (M...m)A) ∈ ℝ)
77	33	r19.20si 1698	. . . . . . . . . . . 12 ⊢ (∀k ∈ (M...m)A ∈ ℂ → ∀k ∈ (M...m)(abs ‘A) ∈ ℝ)
78	56, 77	syl 10	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → ∀k ∈ (M...m)(abs ‘A) ∈ ℝ)
79		fsumreclt 6955	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)(abs ‘A) ∈ ℝ) → Σk ∈ (M...m)(abs ‘A) ∈ ℝ)
80	78, 79	syldan 467	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...m)(abs ‘A) ∈ ℝ)
81		ra4csbela 2032	. . . . . . . . . . 11 ⊢ (((m + 1) ∈ (M...(m + 1)) ⋀ ∀k ∈ (M...(m + 1))(abs ‘A) ∈ ℝ) → [(m + 1) / k](abs ‘A) ∈ ℝ)
82	33	r19.20si 1698	. . . . . . . . . . 11 ⊢ (∀k ∈ (M...(m + 1))A ∈ ℂ → ∀k ∈ (M...(m + 1))(abs ‘A) ∈ ℝ)
83	81, 62, 82	syl2an 454	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → [(m + 1) / k](abs ‘A) ∈ ℝ)
84	74, 76, 80, 83	syl3anc 856	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → ((abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A) ↔ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ≤ (Σk ∈ (M...m)(abs ‘A) + [(m + 1) / k](abs ‘A))))
85	84	biimpa 416	. . . . . . . 8 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) ⋀ (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ≤ (Σk ∈ (M...m)(abs ‘A) + [(m + 1) / k](abs ‘A)))
86		fsump1s 6951	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(abs ‘A) ∈ ℂ) → Σk ∈ (M...(m + 1))(abs ‘A) = (Σk ∈ (M...m)(abs ‘A) + [(m + 1) / k](abs ‘A)))
87	42	r19.20si 1698	. . . . . . . . . 10 ⊢ (∀k ∈ (M...(m + 1))A ∈ ℂ → ∀k ∈ (M...(m + 1))(abs ‘A) ∈ ℂ)
88	86, 87	sylan2 451	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...(m + 1))(abs ‘A) = (Σk ∈ (M...m)(abs ‘A) + [(m + 1) / k](abs ‘A)))
89	88	adantr 389	. . . . . . . 8 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) ⋀ (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → Σk ∈ (M...(m + 1))(abs ‘A) = (Σk ∈ (M...m)(abs ‘A) + [(m + 1) / k](abs ‘A)))
90	85, 89	breqtrrd 2631	. . . . . . 7 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) ⋀ (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ≤ Σk ∈ (M...(m + 1))(abs ‘A))
91		letrt 5498	. . . . . . . . 9 ⊢ (((abs ‘Σk ∈ (M...(m + 1))A) ∈ ℝ ⋀ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ∈ ℝ ⋀ Σk ∈ (M...(m + 1))(abs ‘A) ∈ ℝ) → (((abs ‘Σk ∈ (M...(m + 1))A) ≤ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ⋀ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ≤ Σk ∈ (M...(m + 1))(abs ‘A)) → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A)))
92		fsumclt 6953	. . . . . . . . . . 11 ⊢ (((m + 1) ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...(m + 1))A ∈ ℂ)
93	92, 60	sylan 448	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...(m + 1))A ∈ ℂ)
94		absclt 6768	. . . . . . . . . 10 ⊢ (Σk ∈ (M...(m + 1))A ∈ ℂ → (abs ‘Σk ∈ (M...(m + 1))A) ∈ ℝ)
95	93, 94	syl 10	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → (abs ‘Σk ∈ (M...(m + 1))A) ∈ ℝ)
96		axaddrcl 5244	. . . . . . . . . 10 ⊢ (((abs ‘Σk ∈ (M...m)A) ∈ ℝ ⋀ [(m + 1) / k](abs ‘A) ∈ ℝ) → ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ∈ ℝ)
97	96, 76, 83	sylanc 471	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ∈ ℝ)
98		fsumreclt 6955	. . . . . . . . . 10 ⊢ (((m + 1) ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(abs ‘A) ∈ ℝ) → Σk ∈ (M...(m + 1))(abs ‘A) ∈ ℝ)
99	98, 60, 82	syl2an 454	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...(m + 1))(abs ‘A) ∈ ℝ)
100	91, 95, 97, 99	syl3anc 856	. . . . . . . 8 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → (((abs ‘Σk ∈ (M...(m + 1))A) ≤ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ⋀ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ≤ Σk ∈ (M...(m + 1))(abs ‘A)) → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A)))
101	100	adantr 389	. . . . . . 7 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) ⋀ (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → (((abs ‘Σk ∈ (M...(m + 1))A) ≤ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ⋀ ((abs ‘Σk ∈ (M...m)A) + [(m + 1) / k](abs ‘A)) ≤ Σk ∈ (M...(m + 1))(abs ‘A)) → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A)))
102	73, 90, 101	mp2and 701	. . . . . 6 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) ⋀ (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A))
103	102	exp31 376	. . . . 5 ⊢ (m ∈ (ℤ≥ ‘M) → (∀k ∈ (M...(m + 1))A ∈ ℂ → ((abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A) → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A))))
104	103	a2d 13	. . . 4 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...(m + 1))A ∈ ℂ → (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → (∀k ∈ (M...(m + 1))A ∈ ℂ → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A))))
105	54, 104	syld 27	. . 3 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...m)A ∈ ℂ → (abs ‘Σk ∈ (M...m)A) ≤ Σk ∈ (M...m)(abs ‘A)) → (∀k ∈ (M...(m + 1))A ∈ ℂ → (abs ‘Σk ∈ (M...(m + 1))A) ≤ Σk ∈ (M...(m + 1))(abs ‘A))))
106	7, 14, 21, 28, 46, 105	uzind4 6382	. 2 ⊢ (N ∈ (ℤ≥ ‘M) → (∀k ∈ (M...N)A ∈ ℂ → (abs ‘Σk ∈ (M...N)A) ≤ Σk ∈ (M...N)(abs ‘A)))
107	106	imp 350	1 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...N)A ∈ ℂ) → (abs ‘Σk ∈ (M...N)A) ≤ Σk ∈ (M...N)(abs ‘A))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 953   ∈ wcel 955  ∀wral 1637  Vcvv 1802  [csb 1991   ⊆ wss 2037   class class class wbr 2609   ‘cfv 3172  (class class class)co 3948  ℂcc 5204  ℝcr 5205  1c1 5207   + caddc 5209   ≤ cle 5267  ℤcz 5270  ℤ_≥cuz 6349  ...cfz 6399  abscabs 6681  Σcsu 6917

This theorem is referenced by:  fsumabs2mul 6982  iserzabslem 7114  efaddlem19 7298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-n0 6047  df-z 6083  df-seq1 6245  df-shft 6278  df-uz 6350  df-fz 6400  df-seqz 6465  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-sum 6918

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