Step |
Hyp |
Ref |
Expression |
1 |
|
seqfeq4.m |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
2 |
|
seqfeq4.f |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
3 |
|
seqfeq4.cl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
4 |
|
seqfeq4.id |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑥 𝑄 𝑦 ) ) |
5 |
|
fvex |
⊢ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ∈ V |
6 |
|
fvi |
⊢ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ∈ V → ( I ‘ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ) |
7 |
5 6
|
ax-mp |
⊢ ( I ‘ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) |
8 |
|
ovex |
⊢ ( 𝑥 + 𝑦 ) ∈ V |
9 |
|
fvi |
⊢ ( ( 𝑥 + 𝑦 ) ∈ V → ( I ‘ ( 𝑥 + 𝑦 ) ) = ( 𝑥 + 𝑦 ) ) |
10 |
8 9
|
ax-mp |
⊢ ( I ‘ ( 𝑥 + 𝑦 ) ) = ( 𝑥 + 𝑦 ) |
11 |
|
fvi |
⊢ ( 𝑥 ∈ V → ( I ‘ 𝑥 ) = 𝑥 ) |
12 |
11
|
elv |
⊢ ( I ‘ 𝑥 ) = 𝑥 |
13 |
|
fvi |
⊢ ( 𝑦 ∈ V → ( I ‘ 𝑦 ) = 𝑦 ) |
14 |
13
|
elv |
⊢ ( I ‘ 𝑦 ) = 𝑦 |
15 |
12 14
|
oveq12i |
⊢ ( ( I ‘ 𝑥 ) 𝑄 ( I ‘ 𝑦 ) ) = ( 𝑥 𝑄 𝑦 ) |
16 |
4 10 15
|
3eqtr4g |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( I ‘ ( 𝑥 + 𝑦 ) ) = ( ( I ‘ 𝑥 ) 𝑄 ( I ‘ 𝑦 ) ) ) |
17 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
18 |
|
fvi |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ V → ( I ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
19 |
17 18
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( I ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
20 |
3 2 1 16 19
|
seqhomo |
⊢ ( 𝜑 → ( I ‘ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ) = ( seq 𝑀 ( 𝑄 , 𝐹 ) ‘ 𝑁 ) ) |
21 |
7 20
|
eqtr3id |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) = ( seq 𝑀 ( 𝑄 , 𝐹 ) ‘ 𝑁 ) ) |