Metamath Proof Explorer


Theorem metdsle

Description: The distance from a point to a set is bounded by the distance to any member of the set. (Contributed by Mario Carneiro, 5-Sep-2015)

Ref Expression
Hypothesis metdscn.f 𝐹 = ( 𝑥𝑋 ↦ inf ( ran ( 𝑦𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) )
Assertion metdsle ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( 𝐹𝐵 ) ≤ ( 𝐴 𝐷 𝐵 ) )

Proof

Step Hyp Ref Expression
1 metdscn.f 𝐹 = ( 𝑥𝑋 ↦ inf ( ran ( 𝑦𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) )
2 simprr ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → 𝐵𝑋 )
3 simpr ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) → 𝑆𝑋 )
4 3 sselda ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ 𝐴𝑆 ) → 𝐴𝑋 )
5 4 adantrr ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → 𝐴𝑋 )
6 2 5 jca ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( 𝐵𝑋𝐴𝑋 ) )
7 1 metdstri ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐵𝑋𝐴𝑋 ) ) → ( 𝐹𝐵 ) ≤ ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐹𝐴 ) ) )
8 6 7 syldan ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( 𝐹𝐵 ) ≤ ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐹𝐴 ) ) )
9 simpll ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
10 xmetsym ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐵𝑋𝐴𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( 𝐴 𝐷 𝐵 ) )
11 9 2 5 10 syl3anc ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( 𝐵 𝐷 𝐴 ) = ( 𝐴 𝐷 𝐵 ) )
12 1 metds0 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋𝐴𝑆 ) → ( 𝐹𝐴 ) = 0 )
13 12 3expa ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ 𝐴𝑆 ) → ( 𝐹𝐴 ) = 0 )
14 13 adantrr ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( 𝐹𝐴 ) = 0 )
15 11 14 oveq12d ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐹𝐴 ) ) = ( ( 𝐴 𝐷 𝐵 ) +𝑒 0 ) )
16 xmetcl ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
17 9 5 2 16 syl3anc ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
18 17 xaddid1d ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( ( 𝐴 𝐷 𝐵 ) +𝑒 0 ) = ( 𝐴 𝐷 𝐵 ) )
19 15 18 eqtrd ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐹𝐴 ) ) = ( 𝐴 𝐷 𝐵 ) )
20 8 19 breqtrd ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆𝑋 ) ∧ ( 𝐴𝑆𝐵𝑋 ) ) → ( 𝐹𝐵 ) ≤ ( 𝐴 𝐷 𝐵 ) )