Metamath Proof Explorer


Theorem xpsdsfn2

Description: Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Hypotheses xpsds.t 𝑇 = ( 𝑅 ×s 𝑆 )
xpsds.x 𝑋 = ( Base ‘ 𝑅 )
xpsds.y 𝑌 = ( Base ‘ 𝑆 )
xpsds.1 ( 𝜑𝑅𝑉 )
xpsds.2 ( 𝜑𝑆𝑊 )
xpsds.p 𝑃 = ( dist ‘ 𝑇 )
Assertion xpsdsfn2 ( 𝜑𝑃 Fn ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) )

Proof

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 ( 𝜑𝑅𝑉 )
5 xpsds.2 ( 𝜑𝑆𝑊 )
6 xpsds.p 𝑃 = ( dist ‘ 𝑇 )
7 1 2 3 4 5 6 xpsdsfn ( 𝜑𝑃 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 ( 𝜑𝑃 Fn ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) )