Metamath Proof Explorer
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)
|
|
Ref |
Expression |
|
Hypotheses |
caovclg.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝐸 ) |
|
|
caovcld.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐶 ) |
|
|
caovcld.3 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐷 ) |
|
Assertion |
caovcld |
⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
caovclg.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝐸 ) |
2 |
|
caovcld.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐶 ) |
3 |
|
caovcld.3 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐷 ) |
4 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
5 |
1
|
caovclg |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 ) |
6 |
4 2 3 5
|
syl12anc |
⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 ) |