Theorem gsumfzconst 13844

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 1003	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ (ℤ≥‘𝑀))
2		3simpb 1000	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵))
3		oveq2 5982	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
4	3	mpteq1d 4148	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋))
5	4	oveq2d 5990	. . . . 5 ⊢ (𝑤 = 𝑀 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)))
6		oveq1 5981	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑤 − 𝑀) = (𝑀 − 𝑀))
7	6	oveq1d 5989	. . . . . 6 ⊢ (𝑤 = 𝑀 → ((𝑤 − 𝑀) + 1) = ((𝑀 − 𝑀) + 1))
8	7	oveq1d 5989	. . . . 5 ⊢ (𝑤 = 𝑀 → (((𝑤 − 𝑀) + 1) · 𝑋) = (((𝑀 − 𝑀) + 1) · 𝑋))
9	5, 8	eqeq12d 2224	. . . 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 5982	. . . . . . 7 ⊢ (𝑤 = 𝑗 → (𝑀...𝑤) = (𝑀...𝑗))
12	11	mpteq1d 4148	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋))
13	12	oveq2d 5990	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)))
14		oveq1 5981	. . . . . . 7 ⊢ (𝑤 = 𝑗 → (𝑤 − 𝑀) = (𝑗 − 𝑀))
15	14	oveq1d 5989	. . . . . 6 ⊢ (𝑤 = 𝑗 → ((𝑤 − 𝑀) + 1) = ((𝑗 − 𝑀) + 1))
16	15	oveq1d 5989	. . . . 5 ⊢ (𝑤 = 𝑗 → (((𝑤 − 𝑀) + 1) · 𝑋) = (((𝑗 − 𝑀) + 1) · 𝑋))
17	13, 16	eqeq12d 2224	. . . 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 5982	. . . . . . 7 ⊢ (𝑤 = (𝑗 + 1) → (𝑀...𝑤) = (𝑀...(𝑗 + 1)))
20	19	mpteq1d 4148	. . . . . 6 ⊢ (𝑤 = (𝑗 + 1) → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋))
21	20	oveq2d 5990	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)))
22		oveq1 5981	. . . . . . 7 ⊢ (𝑤 = (𝑗 + 1) → (𝑤 − 𝑀) = ((𝑗 + 1) − 𝑀))
23	22	oveq1d 5989	. . . . . 6 ⊢ (𝑤 = (𝑗 + 1) → ((𝑤 − 𝑀) + 1) = (((𝑗 + 1) − 𝑀) + 1))
24	23	oveq1d 5989	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (((𝑤 − 𝑀) + 1) · 𝑋) = ((((𝑗 + 1) − 𝑀) + 1) · 𝑋))
25	21, 24	eqeq12d 2224	. . . 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 5982	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
28	27	mpteq1d 4148	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋))
29	28	oveq2d 5990	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑤) ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)))
30		oveq1 5981	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 𝑀) = (𝑁 − 𝑀))
31	30	oveq1d 5989	. . . . . 6 ⊢ (𝑤 = 𝑁 → ((𝑤 − 𝑀) + 1) = ((𝑁 − 𝑀) + 1))
32	31	oveq1d 5989	. . . . 5 ⊢ (𝑤 = 𝑁 → (((𝑤 − 𝑀) + 1) · 𝑋) = (((𝑁 − 𝑀) + 1) · 𝑋))
33	29, 32	eqeq12d 2224	. . . 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 528	. . . . . 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 13632	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋)
39	35, 38	syl 14	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (1 · 𝑋) = 𝑋)
40		zcn 9419	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
41	40	subidd 8413	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 𝑀) = 0)
42	41	oveq1d 5989	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 𝑀) + 1) = (0 + 1))
43		0p1e1 9192	. . . . . . . 8 ⊢ (0 + 1) = 1
44	42, 43	eqtrdi 2258	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 𝑀) + 1) = 1)
45	44	oveq1d 5989	. . . . . 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 2209	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
48		simpll 527	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → 𝐺 ∈ Mnd)
49		uzid 9704	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
50	49	adantl 277	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ≥‘𝑀))
51		simpllr 534	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑀)) → 𝑋 ∈ 𝐵)
52	51	fmpttd 5763	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋):(𝑀...𝑀)⟶𝐵)
53	36, 47, 48, 50, 52	gsumval2 13396	. . . . . 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 10620	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) ∈ Fin)
56	55	mptexd 5839	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋) ∈ V)
57		plusgslid 13111	. . . . . . . . 9 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
58	57	slotex 13025	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (+g‘𝐺) ∈ V)
59	48, 58	syl 14	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (+g‘𝐺) ∈ V)
60		seq1g 10652	. . . . . . 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 1252	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋))‘𝑀) = ((𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)‘𝑀))
62		eqid 2209	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋) = (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)
63		eqidd 2210	. . . . . . 7 ⊢ (𝑘 = 𝑀 → 𝑋 = 𝑋)
64		elfz3 10198	. . . . . . . 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 5701	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)‘𝑀) = 𝑋)
67	53, 61, 66	3eqtrd 2246	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑀 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑀) ↦ 𝑋)) = 𝑋)
68	39, 46, 67	3eqtr4rd 2253	. . . 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 10231	. . . . . . . . . . . . 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 5032	. . . . . . . . . . 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 5990	. . . . . . . . . 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 2242	. . . . . . . . 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 2209	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋) = (𝑘 ∈ (𝑀...(𝑗 + 1)) ↦ 𝑋)
77		eqidd 2210	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → 𝑋 = 𝑋)
78		simplr 528	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → 𝑗 ∈ (ℤ≥‘𝑀))
79		peano2uz 9746	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
80		eluzfz2 10196	. . . . . . . . . . 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 534	. . . . . . . . . 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 5701	. . . . . . . . 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 5992	. . . . . . . 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 533	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ (𝐺 Σg (𝑘 ∈ (𝑀...𝑗) ↦ 𝑋)) = (((𝑗 − 𝑀) + 1) · 𝑋)) → 𝐺 ∈ Mnd)
86		eluzel2 9695	. . . . . . . . . 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 534	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑗 + 1))) → 𝑋 ∈ 𝐵)
89	88	fmpttd 5763	. . . . . . . . . 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 13397	. . . . . . . 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 9722	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 − 𝑀) ∈ ℕ0)
93		nn0p1nn 9376	. . . . . . . . . 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 13633	. . . . . . . . 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 2252	. . . . . . 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 9701	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℂ)
99	86	zcnd 9538	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℂ)
100	99	negcld 8412	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ ℂ)
101		1cnd 8130	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 1 ∈ ℂ)
102	98, 100, 101	add32d 8282	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 + -𝑀) + 1) = ((𝑗 + 1) + -𝑀))
103	98, 99	negsubd 8431	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + -𝑀) = (𝑗 − 𝑀))
104	103	oveq1d 5989	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 + -𝑀) + 1) = ((𝑗 − 𝑀) + 1))
105	98, 101	addcld 8134	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ ℂ)
106	105, 99	negsubd 8431	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 + 1) + -𝑀) = ((𝑗 + 1) − 𝑀))
107	102, 104, 106	3eqtr3d 2250	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝑗 − 𝑀) + 1) = ((𝑗 + 1) − 𝑀))
108	107	oveq1d 5989	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (((𝑗 − 𝑀) + 1) + 1) = (((𝑗 + 1) − 𝑀) + 1))
109	108	oveq1d 5989	. . . . . . . 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 2242	. . . . . 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 9751	. 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 983   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ⊆ wss 3177   ↦ cmpt 4124   ↾ cres 4698  ⟶wf 5290  ‘cfv 5294  (class class class)co 5974  Fincfn 6857  0cc0 7967  1c1 7968   + caddc 7970   − cmin 8285  -cneg 8286  ℕcn 9078  ℕ₀cn0 9337  ℤcz 9414  ℤ_≥cuz 9690  ...cfz 10172  seqcseq 10636  Basecbs 12998  +_gcplusg 13076   Σ_g cgsu 13256  Mndcmnd 13415  ._gcmg 13622

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-1o 6532  df-er 6650  df-en 6858  df-fin 6860  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-n0 9338  df-z 9415  df-uz 9691  df-fz 10173  df-seqfrec 10637  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-0g 13257  df-igsum 13258  df-minusg 13503  df-mulg 13623

This theorem is referenced by:  gsumfzconstf  13845  lgseisenlem4  15717