Metamath Proof Explorer


Theorem spanss

Description: Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion spanss ( ( 𝐵 ⊆ ℋ ∧ 𝐴𝐵 ) → ( span ‘ 𝐴 ) ⊆ ( span ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 sstr2 ( 𝐴𝐵 → ( 𝐵𝑥𝐴𝑥 ) )
2 1 adantr ( ( 𝐴𝐵𝑥S ) → ( 𝐵𝑥𝐴𝑥 ) )
3 2 ss2rabdv ( 𝐴𝐵 → { 𝑥S𝐵𝑥 } ⊆ { 𝑥S𝐴𝑥 } )
4 intss ( { 𝑥S𝐵𝑥 } ⊆ { 𝑥S𝐴𝑥 } → { 𝑥S𝐴𝑥 } ⊆ { 𝑥S𝐵𝑥 } )
5 3 4 syl ( 𝐴𝐵 { 𝑥S𝐴𝑥 } ⊆ { 𝑥S𝐵𝑥 } )
6 5 adantl ( ( 𝐵 ⊆ ℋ ∧ 𝐴𝐵 ) → { 𝑥S𝐴𝑥 } ⊆ { 𝑥S𝐵𝑥 } )
7 sstr ( ( 𝐴𝐵𝐵 ⊆ ℋ ) → 𝐴 ⊆ ℋ )
8 7 ancoms ( ( 𝐵 ⊆ ℋ ∧ 𝐴𝐵 ) → 𝐴 ⊆ ℋ )
9 spanval ( 𝐴 ⊆ ℋ → ( span ‘ 𝐴 ) = { 𝑥S𝐴𝑥 } )
10 8 9 syl ( ( 𝐵 ⊆ ℋ ∧ 𝐴𝐵 ) → ( span ‘ 𝐴 ) = { 𝑥S𝐴𝑥 } )
11 spanval ( 𝐵 ⊆ ℋ → ( span ‘ 𝐵 ) = { 𝑥S𝐵𝑥 } )
12 11 adantr ( ( 𝐵 ⊆ ℋ ∧ 𝐴𝐵 ) → ( span ‘ 𝐵 ) = { 𝑥S𝐵𝑥 } )
13 6 10 12 3sstr4d ( ( 𝐵 ⊆ ℋ ∧ 𝐴𝐵 ) → ( span ‘ 𝐴 ) ⊆ ( span ‘ 𝐵 ) )