Theorem gsumfzfsumlemm 14075

Description: Lemma for gsumfzfsum 14076. The case where the sum is inhabited. (Contributed by Jim Kingdon, 9-Sep-2025.)

Hypotheses

Ref	Expression
gsumfzfsumlemm.n	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
gsumfzfsumlemm.b	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
gsumfzfsumlemm	⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)

Distinct variable groups:   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem gsumfzfsumlemm

Dummy variables 𝑗 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumfzfsumlemm.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 10098	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		oveq2 5926	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
5	4	mpteq1d 4114	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵))
6	5	oveq2d 5934	. . . . 5 ⊢ (𝑤 = 𝑀 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)))
7	4	sumeq1d 11509	. . . . 5 ⊢ (𝑤 = 𝑀 → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵)
8	6, 7	eqeq12d 2208	. . . 4 ⊢ (𝑤 = 𝑀 → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵 ↔ (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑀)𝐵))
9	8	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑀 → ((𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵) ↔ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑀)𝐵)))
10		oveq2 5926	. . . . . . 7 ⊢ (𝑤 = 𝑗 → (𝑀...𝑤) = (𝑀...𝑗))
11	10	mpteq1d 4114	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵))
12	11	oveq2d 5934	. . . . 5 ⊢ (𝑤 = 𝑗 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)))
13	10	sumeq1d 11509	. . . . 5 ⊢ (𝑤 = 𝑗 → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...𝑗)𝐵)
14	12, 13	eqeq12d 2208	. . . 4 ⊢ (𝑤 = 𝑗 → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵 ↔ (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵))
15	14	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑗 → ((𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵) ↔ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵)))
16		oveq2 5926	. . . . . . 7 ⊢ (𝑤 = (𝑗 + 1) → (𝑀...𝑤) = (𝑀...(𝑗 + 1)))
17	16	mpteq1d 4114	. . . . . 6 ⊢ (𝑤 = (𝑗 + 1) → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵))
18	17	oveq2d 5934	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)))
19	16	sumeq1d 11509	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵)
20	18, 19	eqeq12d 2208	. . . 4 ⊢ (𝑤 = (𝑗 + 1) → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵 ↔ (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵))
21	20	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑗 + 1) → ((𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵) ↔ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵)))
22		oveq2 5926	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
23	22	mpteq1d 4114	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵))
24	23	oveq2d 5934	. . . . 5 ⊢ (𝑤 = 𝑁 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)))
25	22	sumeq1d 11509	. . . . 5 ⊢ (𝑤 = 𝑁 → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
26	24, 25	eqeq12d 2208	. . . 4 ⊢ (𝑤 = 𝑁 → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵 ↔ (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵))
27	26	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑤)𝐵) ↔ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)))
28		cnfldbas 14051	. . . . . 6 ⊢ ℂ = (Base‘ℂfld)
29		cnring 14058	. . . . . . 7 ⊢ ℂfld ∈ Ring
30		ringmnd 13502	. . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd)
31	29, 30	mp1i 10	. . . . . 6 ⊢ (𝜑 → ℂfld ∈ Mnd)
32		eluzel2 9597	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
33	1, 32	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
34		eluzfz1 10097	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
35	1, 34	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
36		gsumfzfsumlemm.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ)
37	36	ralrimiva 2567	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐵 ∈ ℂ)
38		nfcsb1v 3113	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐵
39	38	nfel1 2347	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐵 ∈ ℂ
40		csbeq1a 3089	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → 𝐵 = ⦋𝑀 / 𝑘⦌𝐵)
41	40	eleq1d 2262	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐵 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐵 ∈ ℂ))
42	39, 41	rspc 2858	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)𝐵 ∈ ℂ → ⦋𝑀 / 𝑘⦌𝐵 ∈ ℂ))
43	35, 37, 42	sylc 62	. . . . . 6 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐵 ∈ ℂ)
44	40	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐵 = ⦋𝑀 / 𝑘⦌𝐵)
45		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑘𝜑
46	28, 31, 33, 43, 44, 45, 38	gsumfzsnfd 13415	. . . . 5 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ {𝑀} ↦ 𝐵)) = ⦋𝑀 / 𝑘⦌𝐵)
47		fzsn 10132	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
48	33, 47	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
49	48	mpteq1d 4114	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵) = (𝑘 ∈ {𝑀} ↦ 𝐵))
50	49	oveq2d 5934	. . . . 5 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ {𝑀} ↦ 𝐵)))
51	47	sumeq1d 11509	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ {𝑀}𝐵)
52	33, 51	syl 14	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ {𝑀}𝐵)
53		sumsns 11558	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ⦋𝑀 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐵 = ⦋𝑀 / 𝑘⦌𝐵)
54	33, 43, 53	syl2anc 411	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐵 = ⦋𝑀 / 𝑘⦌𝐵)
55	52, 54	eqtrd 2226	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 = ⦋𝑀 / 𝑘⦌𝐵)
56	46, 50, 55	3eqtr4d 2236	. . . 4 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑀)𝐵)
57	56	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑀)𝐵))
58		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵) → (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵)
59	58	oveq1d 5933	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵) → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵) = (Σ𝑘 ∈ (𝑀...𝑗)𝐵 + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
60		mpocnfldadd 14053	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
61	29	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ℂfld ∈ Ring)
62	33	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
63		elfzouz 10217	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (ℤ≥‘𝑀))
64	63	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (ℤ≥‘𝑀))
65		simpll 527	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝜑)
66	65, 33	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑀 ∈ ℤ)
67		elfzoel2 10212	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ)
68	67	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑁 ∈ ℤ)
69		elfzelz 10091	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ∈ ℤ)
70	69	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ∈ ℤ)
71		elfzle1 10093	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑀 ≤ 𝑘)
72	71	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑀 ≤ 𝑘)
73	70	zred 9439	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ∈ ℝ)
74		elfzoelz 10213	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
75	74	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑗 ∈ ℤ)
76	75	peano2zd 9442	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝑗 + 1) ∈ ℤ)
77	76	zred 9439	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝑗 + 1) ∈ ℝ)
78	68	zred 9439	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑁 ∈ ℝ)
79		elfzle2 10094	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ≤ (𝑗 + 1))
80	79	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ≤ (𝑗 + 1))
81		fzofzp1 10294	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁))
82	81	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝑗 + 1) ∈ (𝑀...𝑁))
83		elfzle2 10094	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 + 1) ∈ (𝑀...𝑁) → (𝑗 + 1) ≤ 𝑁)
84	82, 83	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝑗 + 1) ≤ 𝑁)
85	73, 77, 78, 80, 84	letrd 8143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ≤ 𝑁)
86	66, 68, 70, 72, 85	elfzd 10082	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ∈ (𝑀...𝑁))
87	65, 86, 36	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝐵 ∈ ℂ)
88	87	fmpttd 5713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵):(𝑀...(𝑗 + 1))⟶ℂ)
89	28, 60, 61, 62, 64, 88	gsumsplit1r 12981	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = ((ℂfld Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) ↾ (𝑀...𝑗)))(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)‘(𝑗 + 1))))
90		fzssp1 10133	. . . . . . . . . . . 12 ⊢ (𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1))
91		resmpt 4990	. . . . . . . . . . . 12 ⊢ ((𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1)) → ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵))
92	90, 91	mp1i 10	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵))
93	92	oveq2d 5934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (ℂfld Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) ↾ (𝑀...𝑗))) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)))
94		peano2uz 9648	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
95	63, 94	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
96	95	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
97		eluzfz2 10098	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (𝑀...(𝑗 + 1)))
98	96, 97	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (𝑀...(𝑗 + 1)))
99		rspcsbela 3140	. . . . . . . . . . . 12 ⊢ (((𝑗 + 1) ∈ (𝑀...𝑁) ∧ ∀𝑘 ∈ (𝑀...𝑁)𝐵 ∈ ℂ) → ⦋(𝑗 + 1) / 𝑘⦌𝐵 ∈ ℂ)
100	81, 37, 99	syl2anr 290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ⦋(𝑗 + 1) / 𝑘⦌𝐵 ∈ ℂ)
101		eqid 2193	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)
102	101	fvmpts 5635	. . . . . . . . . . 11 ⊢ (((𝑗 + 1) ∈ (𝑀...(𝑗 + 1)) ∧ ⦋(𝑗 + 1) / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)‘(𝑗 + 1)) = ⦋(𝑗 + 1) / 𝑘⦌𝐵)
103	98, 100, 102	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)‘(𝑗 + 1)) = ⦋(𝑗 + 1) / 𝑘⦌𝐵)
104	93, 103	oveq12d 5936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((ℂfld Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) ↾ (𝑀...𝑗)))(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)‘(𝑗 + 1))) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵))(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⦋(𝑗 + 1) / 𝑘⦌𝐵))
105		cnfld0 14059	. . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld)
106	29, 30	mp1i 10	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ℂfld ∈ Mnd)
107	74	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
108		fzelp1 10140	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (𝑀...(𝑗 + 1)))
109	108, 87	sylan2 286	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝐵 ∈ ℂ)
110	109	fmpttd 5713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵):(𝑀...𝑗)⟶ℂ)
111	28, 105, 106, 62, 107, 110	gsumfzcl 13071	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) ∈ ℂ)
112	111, 100	addcld 8039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵) ∈ ℂ)
113		oveq1 5925	. . . . . . . . . . 11 ⊢ (𝑥 = (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) → (𝑥 + 𝑦) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + 𝑦))
114		oveq2 5926	. . . . . . . . . . 11 ⊢ (𝑦 = ⦋(𝑗 + 1) / 𝑘⦌𝐵 → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + 𝑦) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
115		eqid 2193	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))
116	113, 114, 115	ovmpog 6053	. . . . . . . . . 10 ⊢ (((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) ∈ ℂ ∧ ⦋(𝑗 + 1) / 𝑘⦌𝐵 ∈ ℂ ∧ ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵) ∈ ℂ) → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵))(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⦋(𝑗 + 1) / 𝑘⦌𝐵) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
117	111, 100, 112, 116	syl3anc 1249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵))(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⦋(𝑗 + 1) / 𝑘⦌𝐵) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
118	89, 104, 117	3eqtrd 2230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
119	118	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵) → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
120		fzsuc 10135	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑗 + 1)) = ((𝑀...𝑗) ∪ {(𝑗 + 1)}))
121	64, 120	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑀...(𝑗 + 1)) = ((𝑀...𝑗) ∪ {(𝑗 + 1)}))
122	121	sumeq1d 11509	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵 = Σ𝑘 ∈ ((𝑀...𝑗) ∪ {(𝑗 + 1)})𝐵)
123		nfv 1539	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁))
124		nfcsb1v 3113	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝑗 + 1) / 𝑘⦌𝐵
125	62, 107	fzfigd 10502	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑀...𝑗) ∈ Fin)
126	107	peano2zd 9442	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
127		fzp1nel 10170	. . . . . . . . . . 11 ⊢ ¬ (𝑗 + 1) ∈ (𝑀...𝑗)
128	127	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ¬ (𝑗 + 1) ∈ (𝑀...𝑗))
129		csbeq1a 3089	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → 𝐵 = ⦋(𝑗 + 1) / 𝑘⦌𝐵)
130	123, 124, 125, 126, 128, 109, 129, 100	fsumsplitsn 11553	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑘 ∈ ((𝑀...𝑗) ∪ {(𝑗 + 1)})𝐵 = (Σ𝑘 ∈ (𝑀...𝑗)𝐵 + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
131	122, 130	eqtrd 2226	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵 = (Σ𝑘 ∈ (𝑀...𝑗)𝐵 + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
132	131	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵) → Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵 = (Σ𝑘 ∈ (𝑀...𝑗)𝐵 + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
133	59, 119, 132	3eqtr4d 2236	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵) → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵)
134	133	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵 → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵))
135	134	expcom 116	. . . 4 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝜑 → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵 → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵)))
136	135	a2d 26	. . 3 ⊢ (𝑗 ∈ (𝑀..^𝑁) → ((𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑗)𝐵) → (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵)))
137	9, 15, 21, 27, 57, 136	fzind2 10306	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵))
138	3, 137	mpcom 36	1 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ⦋csb 3080   ∪ cun 3151   ⊆ wss 3153  {csn 3618   class class class wbr 4029   ↦ cmpt 4090   ↾ cres 4661  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  ℂcc 7870  0cc0 7872  1c1 7873   + caddc 7875   ≤ cle 8055  ℤcz 9317  ℤ_≥cuz 9592  ...cfz 10074  ..^cfzo 10208  Σcsu 11496   Σ_g cgsu 12868  Mndcmnd 12997  Ringcrg 13492  ℂ_fldccnfld 14047

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992  ax-addf 7994  ax-mulf 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-z 9318  df-dec 9449  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497  df-struct 12620  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-starv 12710  df-0g 12869  df-igsum 12870  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-mulg 13190  df-cmn 13356  df-mgp 13417  df-ring 13494  df-cring 13495  df-icnfld 14048

This theorem is referenced by:  gsumfzfsum  14076