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 ) ) )