Theorem rrncms

Description: Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 13-Sep-2015)

	Ref	Expression
Hypothesis	rrncms.1	⊢ 𝑋 = ( ℝ ↑m 𝐼 )
Assertion	rrncms	⊢ ( 𝐼 ∈ Fin → ( ℝn ‘ 𝐼 ) ∈ ( CMet ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		rrncms.1	⊢ 𝑋 = ( ℝ ↑m 𝐼 )
2		eqid	⊢ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) = ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) )
3		eqid	⊢ ( MetOpen ‘ ( ℝn ‘ 𝐼 ) ) = ( MetOpen ‘ ( ℝn ‘ 𝐼 ) )
4		simpll	⊢ ( ( ( 𝐼 ∈ Fin ∧ 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) ) ∧ 𝑓 : ℕ ⟶ 𝑋 ) → 𝐼 ∈ Fin )
5		simplr	⊢ ( ( ( 𝐼 ∈ Fin ∧ 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) ) ∧ 𝑓 : ℕ ⟶ 𝑋 ) → 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) )
6		simpr	⊢ ( ( ( 𝐼 ∈ Fin ∧ 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) ) ∧ 𝑓 : ℕ ⟶ 𝑋 ) → 𝑓 : ℕ ⟶ 𝑋 )
7		eqid	⊢ ( 𝑚 ∈ 𝐼 ↦ ( ⇝ ‘ ( 𝑡 ∈ ℕ ↦ ( ( 𝑓 ‘ 𝑡 ) ‘ 𝑚 ) ) ) ) = ( 𝑚 ∈ 𝐼 ↦ ( ⇝ ‘ ( 𝑡 ∈ ℕ ↦ ( ( 𝑓 ‘ 𝑡 ) ‘ 𝑚 ) ) ) )
8	1 2 3 4 5 6 7	rrncmslem	⊢ ( ( ( 𝐼 ∈ Fin ∧ 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) ) ∧ 𝑓 : ℕ ⟶ 𝑋 ) → 𝑓 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( ℝn ‘ 𝐼 ) ) ) )
9	8	ex	⊢ ( ( 𝐼 ∈ Fin ∧ 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) ) → ( 𝑓 : ℕ ⟶ 𝑋 → 𝑓 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( ℝn ‘ 𝐼 ) ) ) ) )
10	9	ralrimiva	⊢ ( 𝐼 ∈ Fin → ∀ 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) ( 𝑓 : ℕ ⟶ 𝑋 → 𝑓 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( ℝn ‘ 𝐼 ) ) ) ) )
11		nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )
12		1zzd	⊢ ( 𝐼 ∈ Fin → 1 ∈ ℤ )
13	1	rrnmet	⊢ ( 𝐼 ∈ Fin → ( ℝn ‘ 𝐼 ) ∈ ( Met ‘ 𝑋 ) )
14	11 3 12 13	iscmet3	⊢ ( 𝐼 ∈ Fin → ( ( ℝn ‘ 𝐼 ) ∈ ( CMet ‘ 𝑋 ) ↔ ∀ 𝑓 ∈ ( Cau ‘ ( ℝn ‘ 𝐼 ) ) ( 𝑓 : ℕ ⟶ 𝑋 → 𝑓 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( ℝn ‘ 𝐼 ) ) ) ) ) )
15	10 14	mpbird	⊢ ( 𝐼 ∈ Fin → ( ℝn ‘ 𝐼 ) ∈ ( CMet ‘ 𝑋 ) )