Theorem gsumfzfsumlemm 14572

Description: Lemma for gsumfzfsum 14573. 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 10245	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		oveq2 6018	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
5	4	mpteq1d 4169	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵))
6	5	oveq2d 6026	. . . . 5 ⊢ (𝑤 = 𝑀 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)))
7	4	sumeq1d 11898	. . . . 5 ⊢ (𝑤 = 𝑀 → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵)
8	6, 7	eqeq12d 2244	. . . 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 6018	. . . . . . 7 ⊢ (𝑤 = 𝑗 → (𝑀...𝑤) = (𝑀...𝑗))
11	10	mpteq1d 4169	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵))
12	11	oveq2d 6026	. . . . 5 ⊢ (𝑤 = 𝑗 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)))
13	10	sumeq1d 11898	. . . . 5 ⊢ (𝑤 = 𝑗 → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...𝑗)𝐵)
14	12, 13	eqeq12d 2244	. . . 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 6018	. . . . . . 7 ⊢ (𝑤 = (𝑗 + 1) → (𝑀...𝑤) = (𝑀...(𝑗 + 1)))
17	16	mpteq1d 4169	. . . . . 6 ⊢ (𝑤 = (𝑗 + 1) → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵))
18	17	oveq2d 6026	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)))
19	16	sumeq1d 11898	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵)
20	18, 19	eqeq12d 2244	. . . 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 6018	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
23	22	mpteq1d 4169	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵))
24	23	oveq2d 6026	. . . . 5 ⊢ (𝑤 = 𝑁 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)))
25	22	sumeq1d 11898	. . . . 5 ⊢ (𝑤 = 𝑁 → Σ𝑘 ∈ (𝑀...𝑤)𝐵 = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
26	24, 25	eqeq12d 2244	. . . 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 14545	. . . . . 6 ⊢ ℂ = (Base‘ℂfld)
29		cnring 14555	. . . . . . 7 ⊢ ℂfld ∈ Ring
30		ringmnd 13990	. . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd)
31	29, 30	mp1i 10	. . . . . 6 ⊢ (𝜑 → ℂfld ∈ Mnd)
32		eluzel2 9743	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
33	1, 32	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
34		eluzfz1 10244	. . . . . . . 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 2603	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐵 ∈ ℂ)
38		nfcsb1v 3157	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐵
39	38	nfel1 2383	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐵 ∈ ℂ
40		csbeq1a 3133	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → 𝐵 = ⦋𝑀 / 𝑘⦌𝐵)
41	40	eleq1d 2298	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐵 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐵 ∈ ℂ))
42	39, 41	rspc 2901	. . . . . . 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 1574	. . . . . 6 ⊢ Ⅎ𝑘𝜑
46	28, 31, 33, 43, 44, 45, 38	gsumfzsnfd 13903	. . . . 5 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ {𝑀} ↦ 𝐵)) = ⦋𝑀 / 𝑘⦌𝐵)
47		fzsn 10279	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
48	33, 47	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
49	48	mpteq1d 4169	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵) = (𝑘 ∈ {𝑀} ↦ 𝐵))
50	49	oveq2d 6026	. . . . 5 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ {𝑀} ↦ 𝐵)))
51	47	sumeq1d 11898	. . . . . . 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 11947	. . . . . . 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 2262	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 = ⦋𝑀 / 𝑘⦌𝐵)
56	46, 50, 55	3eqtr4d 2272	. . . 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 6025	. . . . . . 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 14546	. . . . . . . . . 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 10364	. . . . . . . . . . 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 10359	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ)
68	67	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑁 ∈ ℤ)
69		elfzelz 10238	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) → 𝑘 ∈ ℤ)
70	69	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ∈ ℤ)
71		elfzle1 10240	. . . . . . . . . . . . . 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 9585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ∈ ℝ)
74		elfzoelz 10360	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
75	74	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑗 ∈ ℤ)
76	75	peano2zd 9588	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝑗 + 1) ∈ ℤ)
77	76	zred 9585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → (𝑗 + 1) ∈ ℝ)
78	68	zred 9585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑁 ∈ ℝ)
79		elfzle2 10241	. . . . . . . . . . . . . . 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 10450	. . . . . . . . . . . . . . . 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 10241	. . . . . . . . . . . . . . 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 8286	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑘 ≤ 𝑁)
86	66, 68, 70, 72, 85	elfzd 10229	. . . . . . . . . . . 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 5795	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵):(𝑀...(𝑗 + 1))⟶ℂ)
89	28, 60, 61, 62, 64, 88	gsumsplit1r 13452	. . . . . . . . 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 10280	. . . . . . . . . . . 12 ⊢ (𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1))
91		resmpt 5056	. . . . . . . . . . . 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 6026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (ℂfld Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) ↾ (𝑀...𝑗))) = (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)))
94		peano2uz 9795	. . . . . . . . . . . . . 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 10245	. . . . . . . . . . . 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 3184	. . . . . . . . . . . 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 2229	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝐵)
102	101	fvmpts 5717	. . . . . . . . . . 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 6028	. . . . . . . . 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 14556	. . . . . . . . . . 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 10287	. . . . . . . . . . . . 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 5795	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵):(𝑀...𝑗)⟶ℂ)
111	28, 105, 106, 62, 107, 110	gsumfzcl 13553	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) ∈ ℂ)
112	111, 100	addcld 8182	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + ⦋(𝑗 + 1) / 𝑘⦌𝐵) ∈ ℂ)
113		oveq1 6017	. . . . . . . . . . 11 ⊢ (𝑥 = (ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) → (𝑥 + 𝑦) = ((ℂfld Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝐵)) + 𝑦))
114		oveq2 6018	. . . . . . . . . . 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 2229	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))
116	113, 114, 115	ovmpog 6148	. . . . . . . . . 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 1271	. . . . . . . . 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 2266	. . . . . . . 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 10282	. . . . . . . . . . 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 11898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑘 ∈ (𝑀...(𝑗 + 1))𝐵 = Σ𝑘 ∈ ((𝑀...𝑗) ∪ {(𝑗 + 1)})𝐵)
123		nfv 1574	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁))
124		nfcsb1v 3157	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝑗 + 1) / 𝑘⦌𝐵
125	62, 107	fzfigd 10670	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑀...𝑗) ∈ Fin)
126	107	peano2zd 9588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
127		fzp1nel 10317	. . . . . . . . . . 11 ⊢ ¬ (𝑗 + 1) ∈ (𝑀...𝑗)
128	127	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ¬ (𝑗 + 1) ∈ (𝑀...𝑗))
129		csbeq1a 3133	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → 𝐵 = ⦋(𝑗 + 1) / 𝑘⦌𝐵)
130	123, 124, 125, 126, 128, 109, 129, 100	fsumsplitsn 11942	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑘 ∈ ((𝑀...𝑗) ∪ {(𝑗 + 1)})𝐵 = (Σ𝑘 ∈ (𝑀...𝑗)𝐵 + ⦋(𝑗 + 1) / 𝑘⦌𝐵))
131	122, 130	eqtrd 2262	. . . . . . . 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 2272	. . . . . 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 10462	. 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 1395   ∈ wcel 2200  ∀wral 2508  ⦋csb 3124   ∪ cun 3195   ⊆ wss 3197  {csn 3666   class class class wbr 4083   ↦ cmpt 4145   ↾ cres 4722  ‘cfv 5321  (class class class)co 6010   ∈ cmpo 6012  ℂcc 8013  0cc0 8015  1c1 8016   + caddc 8018   ≤ cle 8198  ℤcz 9462  ℤ_≥cuz 9738  ...cfz 10221  ..^cfzo 10355  Σcsu 11885   Σ_g cgsu 13311  Mndcmnd 13470  Ringcrg 13980  ℂ_fldccnfld 14541

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135  ax-addf 8137  ax-mulf 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-oadd 6577  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-exp 10778  df-ihash 11015  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-clim 11811  df-sumdc 11886  df-struct 13055  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-plusg 13144  df-mulr 13145  df-starv 13146  df-tset 13150  df-ple 13151  df-ds 13153  df-unif 13154  df-0g 13312  df-igsum 13313  df-topgen 13314  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558  df-mulg 13678  df-cmn 13844  df-mgp 13905  df-ring 13982  df-cring 13983  df-bl 14531  df-mopn 14532  df-fg 14534  df-metu 14535  df-cnfld 14542

This theorem is referenced by:  gsumfzfsum  14573