Step |
Hyp |
Ref |
Expression |
1 |
|
caovdir.1 |
⊢ 𝐴 ∈ V |
2 |
|
caovdir.2 |
⊢ 𝐵 ∈ V |
3 |
|
caovdir.3 |
⊢ 𝐶 ∈ V |
4 |
|
caovdir.com |
⊢ ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 ) |
5 |
|
caovdir.distr |
⊢ ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐹 ( 𝑥 𝐺 𝑧 ) ) |
6 |
|
caovdl.4 |
⊢ 𝐷 ∈ V |
7 |
|
caovdl.5 |
⊢ 𝐻 ∈ V |
8 |
|
caovdl.ass |
⊢ ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) |
9 |
|
ovex |
⊢ ( 𝐴 𝐺 𝐶 ) ∈ V |
10 |
|
ovex |
⊢ ( 𝐵 𝐺 𝐷 ) ∈ V |
11 |
9 10 7 4 5
|
caovdir |
⊢ ( ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐷 ) ) 𝐺 𝐻 ) = ( ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐻 ) 𝐹 ( ( 𝐵 𝐺 𝐷 ) 𝐺 𝐻 ) ) |
12 |
1 3 7 8
|
caovass |
⊢ ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐻 ) = ( 𝐴 𝐺 ( 𝐶 𝐺 𝐻 ) ) |
13 |
2 6 7 8
|
caovass |
⊢ ( ( 𝐵 𝐺 𝐷 ) 𝐺 𝐻 ) = ( 𝐵 𝐺 ( 𝐷 𝐺 𝐻 ) ) |
14 |
12 13
|
oveq12i |
⊢ ( ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐻 ) 𝐹 ( ( 𝐵 𝐺 𝐷 ) 𝐺 𝐻 ) ) = ( ( 𝐴 𝐺 ( 𝐶 𝐺 𝐻 ) ) 𝐹 ( 𝐵 𝐺 ( 𝐷 𝐺 𝐻 ) ) ) |
15 |
11 14
|
eqtri |
⊢ ( ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐷 ) ) 𝐺 𝐻 ) = ( ( 𝐴 𝐺 ( 𝐶 𝐺 𝐻 ) ) 𝐹 ( 𝐵 𝐺 ( 𝐷 𝐺 𝐻 ) ) ) |