Theorem prdsdsfn

Description: Structure product distance function. (Contributed by Mario Carneiro, 15-Sep-2015)

	Ref	Expression
Hypotheses	prdsbas.p	⊢ 𝑃 = ( 𝑆 Xs 𝑅 )
	prdsbas.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
	prdsbas.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
	prdsbas.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	prdsbas.i	⊢ ( 𝜑 → dom 𝑅 = 𝐼 )
	prdsds.l	⊢ 𝐷 = ( dist ‘ 𝑃 )
Assertion	prdsdsfn	⊢ ( 𝜑 → 𝐷 Fn ( 𝐵 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		prdsbas.p	⊢ 𝑃 = ( 𝑆 Xs 𝑅 )
2		prdsbas.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
3		prdsbas.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
4		prdsbas.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
5		prdsbas.i	⊢ ( 𝜑 → dom 𝑅 = 𝐼 )
6		prdsds.l	⊢ 𝐷 = ( dist ‘ 𝑃 )
7		eqid	⊢ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) )
8		xrltso	⊢ < Or ℝ*
9	8	supex	⊢ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ∈ V
10	7 9	fnmpoi	⊢ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) Fn ( 𝐵 × 𝐵 )
11	1 2 3 4 5 6	prdsds	⊢ ( 𝜑 → 𝐷 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) )
12	11	fneq1d	⊢ ( 𝜑 → ( 𝐷 Fn ( 𝐵 × 𝐵 ) ↔ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) Fn ( 𝐵 × 𝐵 ) ) )
13	10 12	mpbiri	⊢ ( 𝜑 → 𝐷 Fn ( 𝐵 × 𝐵 ) )