Step |
Hyp |
Ref |
Expression |
1 |
|
telgsumfz.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
telgsumfz.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
3 |
|
telgsumfz.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
telgsumfz.n |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
5 |
|
telgsumfz.f |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) 𝐴 ∈ 𝐵 ) |
6 |
|
telgsumfz.l |
⊢ ( 𝑘 = 𝑖 → 𝐴 = 𝐿 ) |
7 |
|
telgsumfz.c |
⊢ ( 𝑘 = ( 𝑖 + 1 ) → 𝐴 = 𝐶 ) |
8 |
|
telgsumfz.d |
⊢ ( 𝑘 = 𝑀 → 𝐴 = 𝐷 ) |
9 |
|
telgsumfz.e |
⊢ ( 𝑘 = ( 𝑁 + 1 ) → 𝐴 = 𝐸 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) |
11 |
6
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) ∧ 𝑘 = 𝑖 ) → 𝐴 = 𝐿 ) |
12 |
10 11
|
csbied |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) → ⦋ 𝑖 / 𝑘 ⦌ 𝐴 = 𝐿 ) |
13 |
12
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐿 = ⦋ 𝑖 / 𝑘 ⦌ 𝐴 ) |
14 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝑖 + 1 ) ∈ V ) |
15 |
7
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) ∧ 𝑘 = ( 𝑖 + 1 ) ) → 𝐴 = 𝐶 ) |
16 |
14 15
|
csbied |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) → ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐴 = 𝐶 ) |
17 |
16
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐶 = ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐴 ) |
18 |
13 17
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐿 − 𝐶 ) = ( ⦋ 𝑖 / 𝑘 ⦌ 𝐴 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐴 ) ) |
19 |
18
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 𝑀 ... 𝑁 ) ↦ ( 𝐿 − 𝐶 ) ) = ( 𝑖 ∈ ( 𝑀 ... 𝑁 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐴 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐴 ) ) ) |
20 |
19
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑁 ) ↦ ( 𝐿 − 𝐶 ) ) ) = ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑁 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐴 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐴 ) ) ) ) |
21 |
1 2 3 4 5
|
telgsumfzs |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑁 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐴 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐴 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐴 − ⦋ ( 𝑁 + 1 ) / 𝑘 ⦌ 𝐴 ) ) |
22 |
4
|
elfvexd |
⊢ ( 𝜑 → 𝑀 ∈ V ) |
23 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑀 ) → 𝐴 = 𝐷 ) |
24 |
22 23
|
csbied |
⊢ ( 𝜑 → ⦋ 𝑀 / 𝑘 ⦌ 𝐴 = 𝐷 ) |
25 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ V ) |
26 |
9
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 = ( 𝑁 + 1 ) ) → 𝐴 = 𝐸 ) |
27 |
25 26
|
csbied |
⊢ ( 𝜑 → ⦋ ( 𝑁 + 1 ) / 𝑘 ⦌ 𝐴 = 𝐸 ) |
28 |
24 27
|
oveq12d |
⊢ ( 𝜑 → ( ⦋ 𝑀 / 𝑘 ⦌ 𝐴 − ⦋ ( 𝑁 + 1 ) / 𝑘 ⦌ 𝐴 ) = ( 𝐷 − 𝐸 ) ) |
29 |
20 21 28
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑁 ) ↦ ( 𝐿 − 𝐶 ) ) ) = ( 𝐷 − 𝐸 ) ) |