Theorem fsumadd 6968

Description: The sum of two finite sums.

Assertion

Ref	Expression
fsumadd	⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...N)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...N)(A + B) = (Σk ∈ (M...N)A + Σk ∈ (M...N)B))

Distinct variable groups:   k,M   k,N

Proof of Theorem fsumadd

Step	Hyp	Ref	Expression
1		opreq2 3960	. . . . 5 ⊢ (j = M → (M...j) = (M...M))
2	1	raleq1d 1786	. . . 4 ⊢ (j = M → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)))
3	1	sumeq1d 6936	. . . . 5 ⊢ (j = M → Σk ∈ (M...j)(A + B) = Σk ∈ (M...M)(A + B))
4	1	sumeq1d 6936	. . . . . 6 ⊢ (j = M → Σk ∈ (M...j)A = Σk ∈ (M...M)A)
5	1	sumeq1d 6936	. . . . . 6 ⊢ (j = M → Σk ∈ (M...j)B = Σk ∈ (M...M)B)
6	4, 5	opreq12d 3969	. . . . 5 ⊢ (j = M → (Σk ∈ (M...j)A + Σk ∈ (M...j)B) = (Σk ∈ (M...M)A + Σk ∈ (M...M)B))
7	3, 6	eqeq12d 1486	. . . 4 ⊢ (j = M → (Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B) ↔ Σk ∈ (M...M)(A + B) = (Σk ∈ (M...M)A + Σk ∈ (M...M)B)))
8	2, 7	imbi12d 625	. . 3 ⊢ (j = M → ((∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B)) ↔ (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...M)(A + B) = (Σk ∈ (M...M)A + Σk ∈ (M...M)B))))
9		opreq2 3960	. . . . 5 ⊢ (j = m → (M...j) = (M...m))
10	9	raleq1d 1786	. . . 4 ⊢ (j = m → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ)))
11	9	sumeq1d 6936	. . . . 5 ⊢ (j = m → Σk ∈ (M...j)(A + B) = Σk ∈ (M...m)(A + B))
12	9	sumeq1d 6936	. . . . . 6 ⊢ (j = m → Σk ∈ (M...j)A = Σk ∈ (M...m)A)
13	9	sumeq1d 6936	. . . . . 6 ⊢ (j = m → Σk ∈ (M...j)B = Σk ∈ (M...m)B)
14	12, 13	opreq12d 3969	. . . . 5 ⊢ (j = m → (Σk ∈ (M...j)A + Σk ∈ (M...j)B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B))
15	11, 14	eqeq12d 1486	. . . 4 ⊢ (j = m → (Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B) ↔ Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)))
16	10, 15	imbi12d 625	. . 3 ⊢ (j = m → ((∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B)) ↔ (∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B))))
17		opreq2 3960	. . . . 5 ⊢ (j = (m + 1) → (M...j) = (M...(m + 1)))
18	17	raleq1d 1786	. . . 4 ⊢ (j = (m + 1) → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)))
19	17	sumeq1d 6936	. . . . 5 ⊢ (j = (m + 1) → Σk ∈ (M...j)(A + B) = Σk ∈ (M...(m + 1))(A + B))
20	17	sumeq1d 6936	. . . . . 6 ⊢ (j = (m + 1) → Σk ∈ (M...j)A = Σk ∈ (M...(m + 1))A)
21	17	sumeq1d 6936	. . . . . 6 ⊢ (j = (m + 1) → Σk ∈ (M...j)B = Σk ∈ (M...(m + 1))B)
22	20, 21	opreq12d 3969	. . . . 5 ⊢ (j = (m + 1) → (Σk ∈ (M...j)A + Σk ∈ (M...j)B) = (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B))
23	19, 22	eqeq12d 1486	. . . 4 ⊢ (j = (m + 1) → (Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B) ↔ Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B)))
24	18, 23	imbi12d 625	. . 3 ⊢ (j = (m + 1) → ((∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B)) ↔ (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B))))
25		opreq2 3960	. . . . 5 ⊢ (j = N → (M...j) = (M...N))
26	25	raleq1d 1786	. . . 4 ⊢ (j = N → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...N)(A ∈ ℂ ⋀ B ∈ ℂ)))
27	25	sumeq1d 6936	. . . . 5 ⊢ (j = N → Σk ∈ (M...j)(A + B) = Σk ∈ (M...N)(A + B))
28	25	sumeq1d 6936	. . . . . 6 ⊢ (j = N → Σk ∈ (M...j)A = Σk ∈ (M...N)A)
29	25	sumeq1d 6936	. . . . . 6 ⊢ (j = N → Σk ∈ (M...j)B = Σk ∈ (M...N)B)
30	28, 29	opreq12d 3969	. . . . 5 ⊢ (j = N → (Σk ∈ (M...j)A + Σk ∈ (M...j)B) = (Σk ∈ (M...N)A + Σk ∈ (M...N)B))
31	27, 30	eqeq12d 1486	. . . 4 ⊢ (j = N → (Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B) ↔ Σk ∈ (M...N)(A + B) = (Σk ∈ (M...N)A + Σk ∈ (M...N)B)))
32	26, 31	imbi12d 625	. . 3 ⊢ (j = N → ((∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...j)(A + B) = (Σk ∈ (M...j)A + Σk ∈ (M...j)B)) ↔ (∀k ∈ (M...N)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...N)(A + B) = (Σk ∈ (M...N)A + Σk ∈ (M...N)B))))
33		csbopr12g 3978	. . . . . 6 ⊢ (M ∈ ℤ → [M / k](A + B) = ([M / k]A + [M / k]B))
34	33	adantr 389	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → [M / k](A + B) = ([M / k]A + [M / k]B))
35		fsum1s 6955	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A + B) ∈ ℂ) → Σk ∈ (M...M)(A + B) = [M / k](A + B))
36		axaddcl 5251	. . . . . . 7 ⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A + B) ∈ ℂ)
37	36	r19.20si 1703	. . . . . 6 ⊢ (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...M)(A + B) ∈ ℂ)
38	35, 37	sylan2 451	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...M)(A + B) = [M / k](A + B))
39		fsum1s 6955	. . . . . . 7 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ ℂ) → Σk ∈ (M...M)A = [M / k]A)
40		pm3.26 319	. . . . . . . 8 ⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → A ∈ ℂ)
41	40	r19.20si 1703	. . . . . . 7 ⊢ (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...M)A ∈ ℂ)
42	39, 41	sylan2 451	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...M)A = [M / k]A)
43		fsum1s 6955	. . . . . . 7 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)B ∈ ℂ) → Σk ∈ (M...M)B = [M / k]B)
44		pm3.27 323	. . . . . . . 8 ⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → B ∈ ℂ)
45	44	r19.20si 1703	. . . . . . 7 ⊢ (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...M)B ∈ ℂ)
46	43, 45	sylan2 451	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...M)B = [M / k]B)
47	42, 46	opreq12d 3969	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → (Σk ∈ (M...M)A + Σk ∈ (M...M)B) = ([M / k]A + [M / k]B))
48	34, 38, 47	3eqtr4d 1514	. . . 4 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...M)(A + B) = (Σk ∈ (M...M)A + Σk ∈ (M...M)B))
49	48	ex 373	. . 3 ⊢ (M ∈ ℤ → (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...M)(A + B) = (Σk ∈ (M...M)A + Σk ∈ (M...M)B)))
50		fzssp1t 6446	. . . . . . . . 9 ⊢ ((M ∈ ℤ ⋀ m ∈ ℤ) → (M...m) ⊆ (M...(m + 1)))
51		eluzel2 6364	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → M ∈ ℤ)
52		eluzelz 6363	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → m ∈ ℤ)
53	50, 51, 52	sylanc 471	. . . . . . . 8 ⊢ (m ∈ (ℤ≥ ‘M) → (M...m) ⊆ (M...(m + 1)))
54	53	sseld 2063	. . . . . . 7 ⊢ (m ∈ (ℤ≥ ‘M) → (k ∈ (M...m) → k ∈ (M...(m + 1))))
55	54	imim1d 28	. . . . . 6 ⊢ (m ∈ (ℤ≥ ‘M) → ((k ∈ (M...(m + 1)) → (A ∈ ℂ ⋀ B ∈ ℂ)) → (k ∈ (M...m) → (A ∈ ℂ ⋀ B ∈ ℂ))))
56	55	r19.20dv2 1708	. . . . 5 ⊢ (m ∈ (ℤ≥ ‘M) → (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ)))
57	56	imim1d 28	. . . 4 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)) → (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B))))
58		opreq1 3959	. . . . . . . 8 ⊢ (Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B) → (Σk ∈ (M...m)(A + B) + [(m + 1) / k](A + B)) = ((Σk ∈ (M...m)A + Σk ∈ (M...m)B) + [(m + 1) / k](A + B)))
59		add4t 5318	. . . . . . . . . 10 ⊢ (((Σk ∈ (M...m)A ∈ ℂ ⋀ Σk ∈ (M...m)B ∈ ℂ) ⋀ ([(m + 1) / k]A ∈ ℂ ⋀ [(m + 1) / k]B ∈ ℂ)) → ((Σk ∈ (M...m)A + Σk ∈ (M...m)B) + ([(m + 1) / k]A + [(m + 1) / k]B)) = ((Σk ∈ (M...m)A + [(m + 1) / k]A) + (Σk ∈ (M...m)B + [(m + 1) / k]B)))
60	56	imp 350	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ))
61		fsumclt 6961	. . . . . . . . . . . . 13 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)A ∈ ℂ) → Σk ∈ (M...m)A ∈ ℂ)
62	40	r19.20si 1703	. . . . . . . . . . . . 13 ⊢ (∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...m)A ∈ ℂ)
63	61, 62	sylan2 451	. . . . . . . . . . . 12 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...m)A ∈ ℂ)
64		fsumclt 6961	. . . . . . . . . . . . 13 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)B ∈ ℂ) → Σk ∈ (M...m)B ∈ ℂ)
65	44	r19.20si 1703	. . . . . . . . . . . . 13 ⊢ (∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...m)B ∈ ℂ)
66	64, 65	sylan2 451	. . . . . . . . . . . 12 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...m)B ∈ ℂ)
67	63, 66	jca 288	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ)) → (Σk ∈ (M...m)A ∈ ℂ ⋀ Σk ∈ (M...m)B ∈ ℂ))
68	60, 67	syldan 467	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → (Σk ∈ (M...m)A ∈ ℂ ⋀ Σk ∈ (M...m)B ∈ ℂ))
69		ra4csbela 2038	. . . . . . . . . . . 12 ⊢ (((m + 1) ∈ (M...(m + 1)) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → [(m + 1) / k]A ∈ ℂ)
70		peano2uz 6387	. . . . . . . . . . . . 13 ⊢ (m ∈ (ℤ≥ ‘M) → (m + 1) ∈ (ℤ≥ ‘M))
71		eluzfz2t 6429	. . . . . . . . . . . . 13 ⊢ ((m + 1) ∈ (ℤ≥ ‘M) → (m + 1) ∈ (M...(m + 1)))
72	70, 71	syl 10	. . . . . . . . . . . 12 ⊢ (m ∈ (ℤ≥ ‘M) → (m + 1) ∈ (M...(m + 1)))
73	40	r19.20si 1703	. . . . . . . . . . . 12 ⊢ (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...(m + 1))A ∈ ℂ)
74	69, 72, 73	syl2an 454	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → [(m + 1) / k]A ∈ ℂ)
75		ra4csbela 2038	. . . . . . . . . . . 12 ⊢ (((m + 1) ∈ (M...(m + 1)) ⋀ ∀k ∈ (M...(m + 1))B ∈ ℂ) → [(m + 1) / k]B ∈ ℂ)
76	44	r19.20si 1703	. . . . . . . . . . . 12 ⊢ (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...(m + 1))B ∈ ℂ)
77	75, 72, 76	syl2an 454	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → [(m + 1) / k]B ∈ ℂ)
78	74, 77	jca 288	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → ([(m + 1) / k]A ∈ ℂ ⋀ [(m + 1) / k]B ∈ ℂ))
79	59, 68, 78	sylanc 471	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → ((Σk ∈ (M...m)A + Σk ∈ (M...m)B) + ([(m + 1) / k]A + [(m + 1) / k]B)) = ((Σk ∈ (M...m)A + [(m + 1) / k]A) + (Σk ∈ (M...m)B + [(m + 1) / k]B)))
80		oprex 3974	. . . . . . . . . . 11 ⊢ (m + 1) ∈ V
81		csbopr12g 3978	. . . . . . . . . . 11 ⊢ ((m + 1) ∈ V → [(m + 1) / k](A + B) = ([(m + 1) / k]A + [(m + 1) / k]B))
82	80, 81	ax-mp 7	. . . . . . . . . 10 ⊢ [(m + 1) / k](A + B) = ([(m + 1) / k]A + [(m + 1) / k]B)
83	82	opreq2i 3963	. . . . . . . . 9 ⊢ ((Σk ∈ (M...m)A + Σk ∈ (M...m)B) + [(m + 1) / k](A + B)) = ((Σk ∈ (M...m)A + Σk ∈ (M...m)B) + ([(m + 1) / k]A + [(m + 1) / k]B))
84	79, 83	syl5eq 1516	. . . . . . . 8 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → ((Σk ∈ (M...m)A + Σk ∈ (M...m)B) + [(m + 1) / k](A + B)) = ((Σk ∈ (M...m)A + [(m + 1) / k]A) + (Σk ∈ (M...m)B + [(m + 1) / k]B)))
85	58, 84	sylan9eqr 1526	. . . . . . 7 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) ⋀ Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)) → (Σk ∈ (M...m)(A + B) + [(m + 1) / k](A + B)) = ((Σk ∈ (M...m)A + [(m + 1) / k]A) + (Σk ∈ (M...m)B + [(m + 1) / k]B)))
86		fsump1s 6959	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A + B) ∈ ℂ) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...m)(A + B) + [(m + 1) / k](A + B)))
87	36	r19.20si 1703	. . . . . . . . 9 ⊢ (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...(m + 1))(A + B) ∈ ℂ)
88	86, 87	sylan2 451	. . . . . . . 8 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...m)(A + B) + [(m + 1) / k](A + B)))
89	88	adantr 389	. . . . . . 7 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) ⋀ Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...m)(A + B) + [(m + 1) / k](A + B)))
90		fsump1s 6959	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → Σk ∈ (M...(m + 1))A = (Σk ∈ (M...m)A + [(m + 1) / k]A))
91	90, 73	sylan2 451	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...(m + 1))A = (Σk ∈ (M...m)A + [(m + 1) / k]A))
92		fsump1s 6959	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))B ∈ ℂ) → Σk ∈ (M...(m + 1))B = (Σk ∈ (M...m)B + [(m + 1) / k]B))
93	92, 76	sylan2 451	. . . . . . . . 9 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...(m + 1))B = (Σk ∈ (M...m)B + [(m + 1) / k]B))
94	91, 93	opreq12d 3969	. . . . . . . 8 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B) = ((Σk ∈ (M...m)A + [(m + 1) / k]A) + (Σk ∈ (M...m)B + [(m + 1) / k]B)))
95	94	adantr 389	. . . . . . 7 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) ⋀ Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)) → (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B) = ((Σk ∈ (M...m)A + [(m + 1) / k]A) + (Σk ∈ (M...m)B + [(m + 1) / k]B)))
96	85, 89, 95	3eqtr4d 1514	. . . . . 6 ⊢ (((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) ⋀ Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B))
97	96	exp31 376	. . . . 5 ⊢ (m ∈ (ℤ≥ ‘M) → (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → (Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B))))
98	97	a2d 13	. . . 4 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)) → (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B))))
99	57, 98	syld 27	. . 3 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...m)(A + B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B)) → (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...(m + 1))(A + B) = (Σk ∈ (M...(m + 1))A + Σk ∈ (M...(m + 1))B))))
100	8, 16, 24, 32, 49, 99	uzind4 6390	. 2 ⊢ (N ∈ (ℤ≥ ‘M) → (∀k ∈ (M...N)(A ∈ ℂ ⋀ B ∈ ℂ) → Σk ∈ (M...N)(A + B) = (Σk ∈ (M...N)A + Σk ∈ (M...N)B)))
101	100	imp 350	1 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...N)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...N)(A + B) = (Σk ∈ (M...N)A + Σk ∈ (M...N)B))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∀wral 1642  Vcvv 1807  [csb 1997   ⊆ wss 2043   ‘cfv 3177  (class class class)co 3954  ℂcc 5212  1c1 5215   + caddc 5217  ℤcz 5278  ℤ_≥cuz 6357  ...cfz 6407  Σcsu 6925

This theorem is referenced by:  fsumcom 6974  binomlem5 7016  fnsmnt 7169  fsum0diaglem2 7200  efaddlem6 7293

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-n 5881  df-n0 6055  df-z 6091  df-seq1 6253  df-shft 6286  df-uz 6358  df-fz 6408  df-seqz 6473  df-sum 6926

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