Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝑜 = ⚬ ∧ 𝑚 = 𝑀 ) → 𝑚 = 𝑀 ) |
2 |
|
oveq |
⊢ ( 𝑜 = ⚬ → ( 𝑥 𝑜 𝑦 ) = ( 𝑥 ⚬ 𝑦 ) ) |
3 |
|
oveq |
⊢ ( 𝑜 = ⚬ → ( 𝑦 𝑜 𝑥 ) = ( 𝑦 ⚬ 𝑥 ) ) |
4 |
2 3
|
eqeq12d |
⊢ ( 𝑜 = ⚬ → ( ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ( 𝑥 ⚬ 𝑦 ) = ( 𝑦 ⚬ 𝑥 ) ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝑜 = ⚬ ∧ 𝑚 = 𝑀 ) → ( ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ( 𝑥 ⚬ 𝑦 ) = ( 𝑦 ⚬ 𝑥 ) ) ) |
6 |
1 5
|
raleqbidv |
⊢ ( ( 𝑜 = ⚬ ∧ 𝑚 = 𝑀 ) → ( ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑀 ( 𝑥 ⚬ 𝑦 ) = ( 𝑦 ⚬ 𝑥 ) ) ) |
7 |
1 6
|
raleqbidv |
⊢ ( ( 𝑜 = ⚬ ∧ 𝑚 = 𝑀 ) → ( ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ( 𝑥 ⚬ 𝑦 ) = ( 𝑦 ⚬ 𝑥 ) ) ) |
8 |
|
df-comlaw |
⊢ comLaw = { 〈 𝑜 , 𝑚 〉 ∣ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) } |
9 |
7 8
|
brabga |
⊢ ( ( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊 ) → ( ⚬ comLaw 𝑀 ↔ ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ( 𝑥 ⚬ 𝑦 ) = ( 𝑦 ⚬ 𝑥 ) ) ) |