Theorem fsumadd 7022

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 3975	. . . . 5 ⊢ (j = M → (M...j) = (M...M))
2	1	raleq1d 1792	. . . 4 ⊢ (j = M → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)))
3	1	sumeq1d 6990	. . . . 5 ⊢ (j = M → Σk ∈ (M...j)(A + B) = Σk ∈ (M...M)(A + B))
4	1	sumeq1d 6990	. . . . . 6 ⊢ (j = M → Σk ∈ (M...j)A = Σk ∈ (M...M)A)
5	1	sumeq1d 6990	. . . . . 6 ⊢ (j = M → Σk ∈ (M...j)B = Σk ∈ (M...M)B)
6	4, 5	opreq12d 3984	. . . . 5 ⊢ (j = M → (Σk ∈ (M...j)A + Σk ∈ (M...j)B) = (Σk ∈ (M...M)A + Σk ∈ (M...M)B))
7	3, 6	eqeq12d 1492	. . . 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 628	. . 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 3975	. . . . 5 ⊢ (j = m → (M...j) = (M...m))
10	9	raleq1d 1792	. . . 4 ⊢ (j = m → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ)))
11	9	sumeq1d 6990	. . . . 5 ⊢ (j = m → Σk ∈ (M...j)(A + B) = Σk ∈ (M...m)(A + B))
12	9	sumeq1d 6990	. . . . . 6 ⊢ (j = m → Σk ∈ (M...j)A = Σk ∈ (M...m)A)
13	9	sumeq1d 6990	. . . . . 6 ⊢ (j = m → Σk ∈ (M...j)B = Σk ∈ (M...m)B)
14	12, 13	opreq12d 3984	. . . . 5 ⊢ (j = m → (Σk ∈ (M...j)A + Σk ∈ (M...j)B) = (Σk ∈ (M...m)A + Σk ∈ (M...m)B))
15	11, 14	eqeq12d 1492	. . . 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 628	. . 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 3975	. . . . 5 ⊢ (j = (m + 1) → (M...j) = (M...(m + 1)))
18	17	raleq1d 1792	. . . 4 ⊢ (j = (m + 1) → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)))
19	17	sumeq1d 6990	. . . . 5 ⊢ (j = (m + 1) → Σk ∈ (M...j)(A + B) = Σk ∈ (M...(m + 1))(A + B))
20	17	sumeq1d 6990	. . . . . 6 ⊢ (j = (m + 1) → Σk ∈ (M...j)A = Σk ∈ (M...(m + 1))A)
21	17	sumeq1d 6990	. . . . . 6 ⊢ (j = (m + 1) → Σk ∈ (M...j)B = Σk ∈ (M...(m + 1))B)
22	20, 21	opreq12d 3984	. . . . 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 1492	. . . 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 628	. . 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 3975	. . . . 5 ⊢ (j = N → (M...j) = (M...N))
26	25	raleq1d 1792	. . . 4 ⊢ (j = N → (∀k ∈ (M...j)(A ∈ ℂ ⋀ B ∈ ℂ) ↔ ∀k ∈ (M...N)(A ∈ ℂ ⋀ B ∈ ℂ)))
27	25	sumeq1d 6990	. . . . 5 ⊢ (j = N → Σk ∈ (M...j)(A + B) = Σk ∈ (M...N)(A + B))
28	25	sumeq1d 6990	. . . . . 6 ⊢ (j = N → Σk ∈ (M...j)A = Σk ∈ (M...N)A)
29	25	sumeq1d 6990	. . . . . 6 ⊢ (j = N → Σk ∈ (M...j)B = Σk ∈ (M...N)B)
30	28, 29	opreq12d 3984	. . . . 5 ⊢ (j = N → (Σk ∈ (M...j)A + Σk ∈ (M...j)B) = (Σk ∈ (M...N)A + Σk ∈ (M...N)B))
31	27, 30	eqeq12d 1492	. . . 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 628	. . 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 3993	. . . . . 6 ⊢ (M ∈ ℤ → [M / k](A + B) = ([M / k]A + [M / k]B))
34	33	adantr 391	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → [M / k](A + B) = ([M / k]A + [M / k]B))
35		fsum1s 7009	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A + B) ∈ ℂ) → Σk ∈ (M...M)(A + B) = [M / k](A + B))
36		axaddcl 5283	. . . . . . 7 ⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A + B) ∈ ℂ)
37	36	r19.20si 1709	. . . . . 6 ⊢ (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...M)(A + B) ∈ ℂ)
38	35, 37	sylan2 453	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...M)(A + B) = [M / k](A + B))
39		fsum1s 7009	. . . . . . 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 1709	. . . . . . 7 ⊢ (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...M)A ∈ ℂ)
42	39, 41	sylan2 453	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...M)A = [M / k]A)
43		fsum1s 7009	. . . . . . 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 1709	. . . . . . 7 ⊢ (∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...M)B ∈ ℂ)
46	43, 45	sylan2 453	. . . . . 6 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...M)B = [M / k]B)
47	42, 46	opreq12d 3984	. . . . 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 1520	. . . 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 6507	. . . . . . . . 9 ⊢ ((M ∈ ℤ ⋀ m ∈ ℤ) → (M...m) ⊆ (M...(m + 1)))
51		eluzel2 6425	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → M ∈ ℤ)
52		eluzelz 6424	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → m ∈ ℤ)
53	50, 51, 52	sylanc 473	. . . . . . . 8 ⊢ (m ∈ (ℤ≥ ‘M) → (M...m) ⊆ (M...(m + 1)))
54	53	sseld 2070	. . . . . . 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 1714	. . . . 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 3974	. . . . . . . 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 5350	. . . . . . . . . 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 7015	. . . . . . . . . . . . 13 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)A ∈ ℂ) → Σk ∈ (M...m)A ∈ ℂ)
62	40	r19.20si 1709	. . . . . . . . . . . . 13 ⊢ (∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...m)A ∈ ℂ)
63	61, 62	sylan2 453	. . . . . . . . . . . 12 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ)) → Σk ∈ (M...m)A ∈ ℂ)
64		fsumclt 7015	. . . . . . . . . . . . 13 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...m)B ∈ ℂ) → Σk ∈ (M...m)B ∈ ℂ)
65	44	r19.20si 1709	. . . . . . . . . . . . 13 ⊢ (∀k ∈ (M...m)(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...m)B ∈ ℂ)
66	64, 65	sylan2 453	. . . . . . . . . . . 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 469	. . . . . . . . . 10 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → (Σk ∈ (M...m)A ∈ ℂ ⋀ Σk ∈ (M...m)B ∈ ℂ))
69		ra4csbela 2045	. . . . . . . . . . . 12 ⊢ (((m + 1) ∈ (M...(m + 1)) ⋀ ∀k ∈ (M...(m + 1))A ∈ ℂ) → [(m + 1) / k]A ∈ ℂ)
70		peano2uz 6448	. . . . . . . . . . . . 13 ⊢ (m ∈ (ℤ≥ ‘M) → (m + 1) ∈ (ℤ≥ ‘M))
71		eluzfz2t 6490	. . . . . . . . . . . . 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 1709	. . . . . . . . . . . 12 ⊢ (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...(m + 1))A ∈ ℂ)
74	69, 72, 73	syl2an 456	. . . . . . . . . . 11 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ)) → [(m + 1) / k]A ∈ ℂ)
75		ra4csbela 2045	. . . . . . . . . . . 12 ⊢ (((m + 1) ∈ (M...(m + 1)) ⋀ ∀k ∈ (M...(m + 1))B ∈ ℂ) → [(m + 1) / k]B ∈ ℂ)
76	44	r19.20si 1709	. . . . . . . . . . . 12 ⊢ (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...(m + 1))B ∈ ℂ)
77	75, 72, 76	syl2an 456	. . . . . . . . . . 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 473	. . . . . . . . 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 3989	. . . . . . . . . . 11 ⊢ (m + 1) ∈ V
81		csbopr12g 3993	. . . . . . . . . . 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 3978	. . . . . . . . 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 1522	. . . . . . . 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 1532	. . . . . . 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 7013	. . . . . . . . 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 1709	. . . . . . . . 9 ⊢ (∀k ∈ (M...(m + 1))(A ∈ ℂ ⋀ B ∈ ℂ) → ∀k ∈ (M...(m + 1))(A + B) ∈ ℂ)
88	86, 87	sylan2 453	. . . . . . . 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 391	. . . . . . 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 7013	. . . . . . . . . 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 453	. . . . . . . . 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 7013	. . . . . . . . . 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 453	. . . . . . . . 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 3984	. . . . . . . 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 391	. . . . . . 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 1520	. . . . . 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 378	. . . . 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) → Σ<