Step |
Hyp |
Ref |
Expression |
1 |
|
rngcco.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
2 |
|
rngcco.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
3 |
|
rngcco.o |
⊢ · = ( comp ‘ 𝐶 ) |
4 |
|
rngcco.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑈 ) |
5 |
|
rngcco.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑈 ) |
6 |
|
rngcco.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑈 ) |
7 |
|
rngcco.f |
⊢ ( 𝜑 → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) |
8 |
|
rngcco.g |
⊢ ( 𝜑 → 𝐺 : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑍 ) ) |
9 |
1 2 3
|
rngccofval |
⊢ ( 𝜑 → · = ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) ) |
10 |
9
|
oveqd |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) = ( 〈 𝑋 , 𝑌 〉 ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) 𝑍 ) ) |
11 |
10
|
oveqd |
⊢ ( 𝜑 → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) = ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) 𝑍 ) 𝐹 ) ) |
12 |
|
eqid |
⊢ ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 ) |
13 |
|
eqid |
⊢ ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝑍 ) = ( Base ‘ 𝑍 ) |
17 |
12 2 13 4 5 6 14 15 16 7 8
|
estrcco |
⊢ ( 𝜑 → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) 𝑍 ) 𝐹 ) = ( 𝐺 ∘ 𝐹 ) ) |
18 |
11 17
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) = ( 𝐺 ∘ 𝐹 ) ) |