Step |
Hyp |
Ref |
Expression |
1 |
|
xpsds.t |
⊢ 𝑇 = ( 𝑅 ×s 𝑆 ) |
2 |
|
xpsds.x |
⊢ 𝑋 = ( Base ‘ 𝑅 ) |
3 |
|
xpsds.y |
⊢ 𝑌 = ( Base ‘ 𝑆 ) |
4 |
|
xpsds.1 |
⊢ ( 𝜑 → 𝑅 ∈ 𝑉 ) |
5 |
|
xpsds.2 |
⊢ ( 𝜑 → 𝑆 ∈ 𝑊 ) |
6 |
|
xpsds.p |
⊢ 𝑃 = ( dist ‘ 𝑇 ) |
7 |
1 2 3 4 5 6
|
xpsdsfn |
⊢ ( 𝜑 → 𝑃 Fn ( ( 𝑋 × 𝑌 ) × ( 𝑋 × 𝑌 ) ) ) |
8 |
1 2 3 4 5
|
xpsbas |
⊢ ( 𝜑 → ( 𝑋 × 𝑌 ) = ( Base ‘ 𝑇 ) ) |
9 |
8
|
sqxpeqd |
⊢ ( 𝜑 → ( ( 𝑋 × 𝑌 ) × ( 𝑋 × 𝑌 ) ) = ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) |
10 |
9
|
fneq2d |
⊢ ( 𝜑 → ( 𝑃 Fn ( ( 𝑋 × 𝑌 ) × ( 𝑋 × 𝑌 ) ) ↔ 𝑃 Fn ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) |
11 |
7 10
|
mpbid |
⊢ ( 𝜑 → 𝑃 Fn ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) |