Step |
Hyp |
Ref |
Expression |
1 |
|
caovdirg.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) ) |
2 |
1
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝐾 ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) ) |
3 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) ) |
4 |
3
|
oveq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐹 𝑦 ) 𝐺 𝑧 ) ) |
5 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐺 𝑧 ) = ( 𝐴 𝐺 𝑧 ) ) |
6 |
5
|
oveq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) ) |
7 |
4 6
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) ↔ ( ( 𝐴 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) ) ) |
8 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) ) |
9 |
8
|
oveq1d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝑧 ) ) |
10 |
|
oveq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 𝐺 𝑧 ) = ( 𝐵 𝐺 𝑧 ) ) |
11 |
10
|
oveq2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝐵 𝐺 𝑧 ) ) ) |
12 |
9 11
|
eqeq12d |
⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) ↔ ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝐵 𝐺 𝑧 ) ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑧 = 𝐶 → ( 𝐴 𝐺 𝑧 ) = ( 𝐴 𝐺 𝐶 ) ) |
15 |
|
oveq2 |
⊢ ( 𝑧 = 𝐶 → ( 𝐵 𝐺 𝑧 ) = ( 𝐵 𝐺 𝐶 ) ) |
16 |
14 15
|
oveq12d |
⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝐵 𝐺 𝑧 ) ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) ) |
17 |
13 16
|
eqeq12d |
⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐺 𝑧 ) 𝐻 ( 𝐵 𝐺 𝑧 ) ) ↔ ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) ) ) |
18 |
7 12 17
|
rspc3v |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝐾 ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) ) ) |
19 |
2 18
|
mpan9 |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) ) |