Metamath Proof Explorer


Theorem ressust

Description: The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018)

Ref Expression
Hypotheses ressust.x 𝑋 = ( Base ‘ 𝑊 )
ressust.t 𝑇 = ( UnifSt ‘ ( 𝑊s 𝐴 ) )
Assertion ressust ( ( 𝑊 ∈ UnifSp ∧ 𝐴𝑋 ) → 𝑇 ∈ ( UnifOn ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ressust.x 𝑋 = ( Base ‘ 𝑊 )
2 ressust.t 𝑇 = ( UnifSt ‘ ( 𝑊s 𝐴 ) )
3 1 fvexi 𝑋 ∈ V
4 3 ssex ( 𝐴𝑋𝐴 ∈ V )
5 4 adantl ( ( 𝑊 ∈ UnifSp ∧ 𝐴𝑋 ) → 𝐴 ∈ V )
6 ressuss ( 𝐴 ∈ V → ( UnifSt ‘ ( 𝑊s 𝐴 ) ) = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) )
7 5 6 syl ( ( 𝑊 ∈ UnifSp ∧ 𝐴𝑋 ) → ( UnifSt ‘ ( 𝑊s 𝐴 ) ) = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) )
8 2 7 syl5eq ( ( 𝑊 ∈ UnifSp ∧ 𝐴𝑋 ) → 𝑇 = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) )
9 eqid ( UnifSt ‘ 𝑊 ) = ( UnifSt ‘ 𝑊 )
10 eqid ( TopOpen ‘ 𝑊 ) = ( TopOpen ‘ 𝑊 )
11 1 9 10 isusp ( 𝑊 ∈ UnifSp ↔ ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝑋 ) ∧ ( TopOpen ‘ 𝑊 ) = ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ) )
12 11 simplbi ( 𝑊 ∈ UnifSp → ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝑋 ) )
13 trust ( ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) → ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) )
14 12 13 sylan ( ( 𝑊 ∈ UnifSp ∧ 𝐴𝑋 ) → ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) )
15 8 14 eqeltrd ( ( 𝑊 ∈ UnifSp ∧ 𝐴𝑋 ) → 𝑇 ∈ ( UnifOn ‘ 𝐴 ) )