Step |
Hyp |
Ref |
Expression |
1 |
|
prodmo.1 |
⊢ 𝐹 = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) |
2 |
|
prodmo.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
3 |
|
prodrb.4 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
4 |
|
prodrb.5 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
5 |
|
prodrb.6 |
⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
6 |
|
prodrb.7 |
⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑁 ) ) |
7 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑁 ∈ ℤ ) |
8 |
|
seqex |
⊢ seq 𝑀 ( · , 𝐹 ) ∈ V |
9 |
|
climres |
⊢ ( ( 𝑁 ∈ ℤ ∧ seq 𝑀 ( · , 𝐹 ) ∈ V ) → ( ( seq 𝑀 ( · , 𝐹 ) ↾ ( ℤ≥ ‘ 𝑁 ) ) ⇝ 𝐶 ↔ seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ) ) |
10 |
7 8 9
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( seq 𝑀 ( · , 𝐹 ) ↾ ( ℤ≥ ‘ 𝑁 ) ) ⇝ 𝐶 ↔ seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ) ) |
11 |
2
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
13 |
1 11 12
|
prodrblem |
⊢ ( ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑁 ) ) → ( seq 𝑀 ( · , 𝐹 ) ↾ ( ℤ≥ ‘ 𝑁 ) ) = seq 𝑁 ( · , 𝐹 ) ) |
14 |
6 13
|
mpidan |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( seq 𝑀 ( · , 𝐹 ) ↾ ( ℤ≥ ‘ 𝑁 ) ) = seq 𝑁 ( · , 𝐹 ) ) |
15 |
14
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( seq 𝑀 ( · , 𝐹 ) ↾ ( ℤ≥ ‘ 𝑁 ) ) ⇝ 𝐶 ↔ seq 𝑁 ( · , 𝐹 ) ⇝ 𝐶 ) ) |
16 |
10 15
|
bitr3d |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ↔ seq 𝑁 ( · , 𝐹 ) ⇝ 𝐶 ) ) |