Step |
Hyp |
Ref |
Expression |
1 |
|
seqp1d.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
seqp1d.2 |
⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) |
3 |
|
seqp1d.3 |
⊢ 𝐾 = ( 𝑁 + 1 ) |
4 |
|
seqp1d.4 |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) = 𝐴 ) |
5 |
|
seqp1d.5 |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐾 ) = 𝐵 ) |
6 |
3
|
fveq2i |
⊢ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑁 + 1 ) ) |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑁 + 1 ) ) ) |
8 |
2 1
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
9 |
|
seqp1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑁 + 1 ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) + ( 𝐹 ‘ ( 𝑁 + 1 ) ) ) ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑁 + 1 ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) + ( 𝐹 ‘ ( 𝑁 + 1 ) ) ) ) |
11 |
3
|
fveq2i |
⊢ ( 𝐹 ‘ 𝐾 ) = ( 𝐹 ‘ ( 𝑁 + 1 ) ) |
12 |
11 5
|
eqtr3id |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝑁 + 1 ) ) = 𝐵 ) |
13 |
4 12
|
oveq12d |
⊢ ( 𝜑 → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) + ( 𝐹 ‘ ( 𝑁 + 1 ) ) ) = ( 𝐴 + 𝐵 ) ) |
14 |
7 10 13
|
3eqtrd |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = ( 𝐴 + 𝐵 ) ) |