cnmpt1ds

Description: Continuity of the metric function; analogue of cnmpt12f which cannot be used directly because D is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	cnmpt1ds.d	⊢ 𝐷 = ( dist ‘ 𝐺 )
	cnmpt1ds.j	⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
	cnmpt1ds.r	⊢ 𝑅 = ( topGen ‘ ran (,) )
	cnmpt1ds.g	⊢ ( 𝜑 → 𝐺 ∈ MetSp )
	cnmpt1ds.k	⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑋 ) )
	cnmpt1ds.a	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝐾 Cn 𝐽 ) )
	cnmpt1ds.b	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ∈ ( 𝐾 Cn 𝐽 ) )
Assertion	cnmpt1ds	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 𝐷 𝐵 ) ) ∈ ( 𝐾 Cn 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		cnmpt1ds.d	⊢ 𝐷 = ( dist ‘ 𝐺 )
2		cnmpt1ds.j	⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
3		cnmpt1ds.r	⊢ 𝑅 = ( topGen ‘ ran (,) )
4		cnmpt1ds.g	⊢ ( 𝜑 → 𝐺 ∈ MetSp )
5		cnmpt1ds.k	⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑋 ) )
6		cnmpt1ds.a	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝐾 Cn 𝐽 ) )
7		cnmpt1ds.b	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ∈ ( 𝐾 Cn 𝐽 ) )
8		mstps	⊢ ( 𝐺 ∈ MetSp → 𝐺 ∈ TopSp )
9	4 8	syl	⊢ ( 𝜑 → 𝐺 ∈ TopSp )
10		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
11	10 2	istps	⊢ ( 𝐺 ∈ TopSp ↔ 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) )
12	9 11	sylib	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) )
13		cnf2	⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) ∧ ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝐾 Cn 𝐽 ) ) → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) : 𝑋 ⟶ ( Base ‘ 𝐺 ) )
14	5 12 6 13	syl3anc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) : 𝑋 ⟶ ( Base ‘ 𝐺 ) )
15	14	fvmptelcdm	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ ( Base ‘ 𝐺 ) )
16		cnf2	⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) ∧ ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ∈ ( 𝐾 Cn 𝐽 ) ) → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) : 𝑋 ⟶ ( Base ‘ 𝐺 ) )
17	5 12 7 16	syl3anc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) : 𝑋 ⟶ ( Base ‘ 𝐺 ) )
18	17	fvmptelcdm	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐵 ∈ ( Base ‘ 𝐺 ) )
19	15 18	ovresd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
20	19	mpteq2dva	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝐵 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 𝐷 𝐵 ) ) )
21	10 1 2 3	msdcn	⊢ ( 𝐺 ∈ MetSp → ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ( 𝐽 ×_t 𝐽 ) Cn 𝑅 ) )
22	4 21	syl	⊢ ( 𝜑 → ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ( 𝐽 ×_t 𝐽 ) Cn 𝑅 ) )
23	5 6 7 22	cnmpt12f	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 ( 𝐷 ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝐵 ) ) ∈ ( 𝐾 Cn 𝑅 ) )
24	20 23	eqeltrrd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 𝐷 𝐵 ) ) ∈ ( 𝐾 Cn 𝑅 ) )

Metamath Proof Explorer

Theorem cnmpt1ds

Proof