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Metamath Proof Explorer

Theorem blssp

Description: A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 5-Jan-2014)

Ref Expression
Hypothesis blssp.2 ⊢ 𝑁 = ( 𝑀 ↾ ( 𝑆 × 𝑆 ) )
Assertion blssp ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → ( 𝑌 ( ball ‘ 𝑁 ) 𝑅 ) = ( ( 𝑌 ( ball ‘ 𝑀 ) 𝑅 ) ∩ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 blssp.2 ⊢ 𝑁 = ( 𝑀 ↾ ( 𝑆 × 𝑆 ) )
2 metxmet ⊢ ( 𝑀 ∈ ( Met ‘ 𝑋 ) → 𝑀 ∈ ( ∞Met ‘ 𝑋 ) )
3 2 ad2antrr ⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → 𝑀 ∈ ( ∞Met ‘ 𝑋 ) )
4 simprl ⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → 𝑌 ∈ 𝑆 )
5 simplr ⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → 𝑆 ⊆ 𝑋 )
6 sseqin2 ⊢ ( 𝑆 ⊆ 𝑋 ↔ ( 𝑋 ∩ 𝑆 ) = 𝑆 )
7 5 6 sylib ⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → ( 𝑋 ∩ 𝑆 ) = 𝑆 )
8 4 7 eleqtrrd ⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → 𝑌 ∈ ( 𝑋 ∩ 𝑆 ) )
9 rpxr ⊢ ( 𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ* )
10 9 ad2antll ⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → 𝑅 ∈ ℝ* )
11 1 blres ⊢ ( ( 𝑀 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑌 ∈ ( 𝑋 ∩ 𝑆 ) ∧ 𝑅 ∈ ℝ* ) → ( 𝑌 ( ball ‘ 𝑁 ) 𝑅 ) = ( ( 𝑌 ( ball ‘ 𝑀 ) 𝑅 ) ∩ 𝑆 ) )
12 3 8 10 11 syl3anc ⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+ ) ) → ( 𝑌 ( ball ‘ 𝑁 ) 𝑅 ) = ( ( 𝑌 ( ball ‘ 𝑀 ) 𝑅 ) ∩ 𝑆 ) )