Step |
Hyp |
Ref |
Expression |
1 |
|
mscl.x |
⊢ 𝑋 = ( Base ‘ 𝑀 ) |
2 |
|
mscl.d |
⊢ 𝐷 = ( dist ‘ 𝑀 ) |
3 |
|
ovres |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) ) |
4 |
3
|
3adant1 |
⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) ) |
5 |
4
|
eqeq1d |
⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = 0 ↔ ( 𝐴 𝐷 𝐵 ) = 0 ) ) |
6 |
1 2
|
xmsxmet2 |
⊢ ( 𝑀 ∈ ∞MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) ) |
7 |
|
xmeteq0 |
⊢ ( ( ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) ) |
8 |
6 7
|
syl3an1 |
⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) ) |
9 |
5 8
|
bitr3d |
⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) ) |