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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 ⊢ ( 𝜑 → 𝑅 ∈ 𝑉 )
xpsds.2 ⊢ ( 𝜑 → 𝑆 ∈ 𝑊 )
xpsds.p ⊢ 𝑃 = ( dist ‘ 𝑇 )
Assertion xpsdsfn2 ( 𝜑 → 𝑃 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 ⊢ ( 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 ) ) )