Theorem xmetlecl

Description: Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xmetlecl	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ ℝ ∧ ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 ) ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		xmetcl	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
2	1	3expb	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
3	2	3adant3	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ ℝ ∧ ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 ) ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
4		simp3l	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ ℝ ∧ ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 ) ) → 𝐶 ∈ ℝ )
5		xmetge0	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 0 ≤ ( 𝐴 𝐷 𝐵 ) )
6	5	3expb	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → 0 ≤ ( 𝐴 𝐷 𝐵 ) )
7	6	3adant3	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ ℝ ∧ ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 ) ) → 0 ≤ ( 𝐴 𝐷 𝐵 ) )
8		simp3r	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ ℝ ∧ ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 )
9		xrrege0	⊢ ( ( ( ( 𝐴 𝐷 𝐵 ) ∈ ℝ* ∧ 𝐶 ∈ ℝ ) ∧ ( 0 ≤ ( 𝐴 𝐷 𝐵 ) ∧ ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 ) ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )
10	3 4 7 8 9	syl22anc	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( 𝐶 ∈ ℝ ∧ ( 𝐴 𝐷 𝐵 ) ≤ 𝐶 ) ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )