Theorem gsumfzconst 14047

Description: Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)

Hypotheses

Ref	Expression
gsumconst.b	⊢ 𝐵 = (Base‘𝐺)
gsumconst.m	⊢ · = (.g‘𝐺)

Assertion

Ref	Expression
gsumfzconst	⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))

Distinct variable groups:   𝐵,𝑘   𝑘,𝐺   𝑘,𝑀   𝑘,𝑁   𝑘,𝑋

Allowed substitution hint:   · (𝑘)

Proof of Theorem gsumfzconst

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

Step	Hyp	Ref	Expression
1		simp2 1025	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ (ℤ≥‘𝑀))
2		3simpb 1022	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵))
3		oveq2 6057	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
4	3	mpteq1d 4194	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋))
5	4	oveq2d 6065	. . . . 5 ⊢ (𝑤 = 𝑀 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)))
6		oveq1 6056	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑤 − 𝑀) = (𝑀 − 𝑀))
7	6	oveq1d 6064	. . . . . 6 ⊢ (𝑤 = 𝑀 → ((𝑤 − 𝑀) + 1) = ((𝑀 − 𝑀) + 1))
8	7	oveq1d 6064	. . . . 5 ⊢ (𝑤 = 𝑀 → (((𝑤 − 𝑀) + 1) · 𝑋) = (((𝑀 − 𝑀) + 1) · 𝑋))
9	5, 8	eqeq12d 2247	. . . 4 ⊢ (𝑤 = 𝑀 → ((𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋) ↔ (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)) = (((𝑀 − 𝑀) + 1) · 𝑋)))
10	9	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑀 → (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋)) ↔ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)) = (((𝑀 − 𝑀) + 1) · 𝑋))))
11		oveq2 6057	. . . . . . 7 ⊢ (𝑤 = 𝑗 → (𝑀...𝑤) = (𝑀...𝑗))
12	11	mpteq1d 4194	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋))
13	12	oveq2d 6065	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)))
14		oveq1 6056	. . . . . . 7 ⊢ (𝑤 = 𝑗 → (𝑤 − 𝑀) = (𝑗 − 𝑀))
15	14	oveq1d 6064	. . . . . 6 ⊢ (𝑤 = 𝑗 → ((𝑤 − 𝑀) + 1) = ((𝑗 − 𝑀) + 1))
16	15	oveq1d 6064	. . . . 5 ⊢ (𝑤 = 𝑗 → (((𝑤 − 𝑀) + 1) · 𝑋) = (((𝑗 − 𝑀) + 1) · 𝑋))
17	13, 16	eqeq12d 2247	. . . 4 ⊢ (𝑤 = 𝑗 → ((𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋) ↔ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)))
18	17	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑗 → (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋)) ↔ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋))))
19		oveq2 6057	. . . . . . 7 ⊢ (𝑤 = (𝑗 + 1) → (𝑀...𝑤) = (𝑀...(𝑗 + 1)))
20	19	mpteq1d 4194	. . . . . 6 ⊢ (𝑤 = (𝑗 + 1) → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋))
21	20	oveq2d 6065	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)))
22		oveq1 6056	. . . . . . 7 ⊢ (𝑤 = (𝑗 + 1) → (𝑤 − 𝑀) = ((𝑗 + 1) − 𝑀))
23	22	oveq1d 6064	. . . . . 6 ⊢ (𝑤 = (𝑗 + 1) → ((𝑤 − 𝑀) + 1) = (((𝑗 + 1) − 𝑀) + 1))
24	23	oveq1d 6064	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (((𝑤 − 𝑀) + 1) · 𝑋) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))
25	21, 24	eqeq12d 2247	. . . 4 ⊢ (𝑤 = (𝑗 + 1) → ((𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋) ↔ (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋)))
26	25	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑗 + 1) → (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋)) ↔ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))))
27		oveq2 6057	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
28	27	mpteq1d 4194	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋))
29	28	oveq2d 6065	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)))
30		oveq1 6056	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 𝑀) = (𝑁 − 𝑀))
31	30	oveq1d 6064	. . . . . 6 ⊢ (𝑤 = 𝑁 → ((𝑤 − 𝑀) + 1) = ((𝑁 − 𝑀) + 1))
32	31	oveq1d 6064	. . . . 5 ⊢ (𝑤 = 𝑁 → (((𝑤 − 𝑀) + 1) · 𝑋) = (((𝑁 − 𝑀) + 1) · 𝑋))
33	29, 32	eqeq12d 2247	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋) ↔ (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋)))
34	33	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (((𝑤 − 𝑀) + 1) · 𝑋)) ↔ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))))
35		simplr 529	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → 𝑋 ∈ 𝐵)
36		gsumconst.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
37		gsumconst.m	. . . . . . 7 ⊢ · = (.g‘𝐺)
38	36, 37	mulg1 13835	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋)
39	35, 38	syl 14	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (1 · 𝑋) = 𝑋)
40		zcn 9578	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
41	40	subidd 8568	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 𝑀) = 0)
42	41	oveq1d 6064	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 𝑀) + 1) = (0 + 1))
43		0p1e1 9347	. . . . . . . 8 ⊢ (0 + 1) = 1
44	42, 43	eqtrdi 2281	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 𝑀) + 1) = 1)
45	44	oveq1d 6064	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (((𝑀 − 𝑀) + 1) · 𝑋) = (1 · 𝑋))
46	45	adantl 277	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (((𝑀 − 𝑀) + 1) · 𝑋) = (1 · 𝑋))
47		eqid 2232	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
48		simpll 527	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → 𝐺 ∈ Mnd)
49		uzid 9864	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
50	49	adantl 277	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ≥‘𝑀))
51		simpllr 536	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑀)) → 𝑋 ∈ 𝐵)
52	51	fmpttd 5831	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋):(𝑀...𝑀)⟶𝐵)
53	36, 47, 48, 50, 52	gsumval2 13599	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)) = (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋))‘𝑀))
54		simpr 110	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
55	54, 54	fzfigd 10789	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) ∈ Fin)
56	55	mptexd 5912	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋) ∈ V)
57		plusgslid 13314	. . . . . . . . 9 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
58	57	slotex 13228	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (+g‘𝐺) ∈ V)
59	48, 58	syl 14	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (+g‘𝐺) ∈ V)
60		seq1g 10821	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋) ∈ V ∧ (+g‘𝐺) ∈ V) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋))‘𝑀) = ((𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)‘𝑀))
61	54, 56, 59, 60	syl3anc 1274	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋))‘𝑀) = ((𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)‘𝑀))
62		eqid 2232	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)
63		eqidd 2233	. . . . . . 7 ⊢ (𝑘 = 𝑀 → 𝑋 = 𝑋)
64		elfz3 10364	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀))
65	64	adantl 277	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ (𝑀...𝑀))
66	62, 63, 65, 35	fvmptd3 5770	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)‘𝑀) = 𝑋)
67	53, 61, 66	3eqtrd 2269	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)) = 𝑋)
68	39, 46, 67	3eqtr4rd 2276	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)) = (((𝑀 − 𝑀) + 1) · 𝑋))
69	68	expcom 116	. . 3 ⊢ (𝑀 ∈ ℤ → ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)) = (((𝑀 − 𝑀) + 1) · 𝑋)))
70		fzssp1 10397	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1))
71	70	a1i 9	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1)))
72	71	resmptd 5088	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋))
73	72	oveq2d 6065	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝐺 Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋) ↾ (𝑀...𝑗))) = (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)))
74		simpr 110	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋))
75	73, 74	eqtrd 2265	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝐺 Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋) ↾ (𝑀...𝑗))) = (((𝑗 − 𝑀) + 1) · 𝑋))
76		eqid 2232	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)
77		eqidd 2233	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → 𝑋 = 𝑋)
78		simplr 529	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → 𝑗 ∈ (ℤ≥‘𝑀))
79		peano2uz 9911	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
80		eluzfz2 10362	. . . . . . . . . . 11 ⊢ ((𝑗 + 1) ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (𝑀...(𝑗 + 1)))
81	78, 79, 80	3syl 17	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝑗 + 1) ∈ (𝑀...(𝑗 + 1)))
82		simpllr 536	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → 𝑋 ∈ 𝐵)
83	76, 77, 81, 82	fvmptd3 5770	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)‘(𝑗 + 1)) = 𝑋)
84	75, 83	oveq12d 6067	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → ((𝐺 Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋) ↾ (𝑀...𝑗)))(+g‘𝐺)((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)‘(𝑗 + 1))) = ((((𝑗 − 𝑀) + 1) · 𝑋)(+g‘𝐺)𝑋))
85		simplll 535	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → 𝐺 ∈ Mnd)
86		eluzel2 9854	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
87	78, 86	syl 14	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → 𝑀 ∈ ℤ)
88		simpllr 536	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑋 ∈ 𝐵)
89	88	fmpttd 5831	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋):(𝑀...(𝑗 + 1))⟶𝐵)
90	89	adantr 276	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋):(𝑀...(𝑗 + 1))⟶𝐵)
91	36, 47, 85, 87, 78, 90	gsumsplit1r 13600	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((𝐺 Σg ((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋) ↾ (𝑀...𝑗)))(+g‘𝐺)((𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)‘(𝑗 + 1))))
92		uznn0sub 9882	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 − 𝑀) ∈ ℕ0)
93		nn0p1nn 9531	. . . . . . . . . 10 ⊢ ((𝑗 − 𝑀) ∈ ℕ0 → ((𝑗 − 𝑀) + 1) ∈ ℕ)
94	78, 92, 93	3syl 17	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → ((𝑗 − 𝑀) + 1) ∈ ℕ)
95	36, 37, 47	mulgnnp1 13836	. . . . . . . . 9 ⊢ ((((𝑗 − 𝑀) + 1) ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((((𝑗 − 𝑀) + 1) + 1) · 𝑋) = ((((𝑗 − 𝑀) + 1) · 𝑋)(+g‘𝐺)𝑋))
96	94, 82, 95	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → ((((𝑗 − 𝑀) + 1) + 1) · 𝑋) = ((((𝑗 − 𝑀) + 1) · 𝑋)(+g‘𝐺)𝑋))
97	84, 91, 96	3eqtr4d 2275	. . . . . . 7 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((((𝑗 − 𝑀) + 1) + 1) · 𝑋))
98		eluzelcn 9861	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℂ)
99	86	zcnd 9697	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℂ)
100	99	negcld 8567	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ ℂ)
101		1cnd 8286	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 1 ∈ ℂ)
102	98, 100, 101	add32d 8437	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 + -𝑀) + 1) = ((𝑗 + 1) + -𝑀))
103	98, 99	negsubd 8586	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + -𝑀) = (𝑗 − 𝑀))
104	103	oveq1d 6064	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 + -𝑀) + 1) = ((𝑗 − 𝑀) + 1))
105	98, 101	addcld 8289	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ ℂ)
106	105, 99	negsubd 8586	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 + 1) + -𝑀) = ((𝑗 + 1) − 𝑀))
107	102, 104, 106	3eqtr3d 2273	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 − 𝑀) + 1) = ((𝑗 + 1) − 𝑀))
108	107	oveq1d 6064	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (((𝑗 − 𝑀) + 1) + 1) = (((𝑗 + 1) − 𝑀) + 1))
109	108	oveq1d 6064	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((((𝑗 − 𝑀) + 1) + 1) · 𝑋) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))
110	78, 109	syl 14	. . . . . . 7 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → ((((𝑗 − 𝑀) + 1) + 1) · 𝑋) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))
111	97, 110	eqtrd 2265	. . . . . 6 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))
112	111	ex 115	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ((𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋) → (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋)))
113	112	expcom 116	. . . 4 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋) → (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))))
114	113	a2d 26	. . 3 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))))
115	10, 18, 26, 34, 69, 114	uzind4 9916	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋)))
116	1, 2, 115	sylc 62	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210   ↦ cmpt 4170   ↾ cres 4750  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  Fincfn 6974  0cc0 8123  1c1 8124   + caddc 8126   − cmin 8440  -cneg 8441  ℕcn 9233  ℕ₀cn0 9492  ℤcz 9573  ℤ_≥cuz 9849  ...cfz 10338  seqcseq 10805  Basecbs 13201  +_gcplusg 13279   Σ_g cgsu 13459  Mndcmnd 13618  ._gcmg 13825

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-en 6975  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-2 9292  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339  df-seqfrec 10806  df-ndx 13204  df-slot 13205  df-base 13207  df-plusg 13292  df-0g 13460  df-igsum 13461  df-minusg 13706  df-mulg 13826

This theorem is referenced by:  gsumfzconstf  14048  lgseisenlem4  15933