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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 C_ ~H /\ A C_ B ) -> ( span ` A ) C_ ( span ` B ) )

Proof

Step Hyp Ref Expression
1 sstr2 
 |-  ( A C_ B -> ( B C_ x -> A C_ x ) )
2 1 adantr 
 |-  ( ( A C_ B /\ x e. SH ) -> ( B C_ x -> A C_ x ) )
3 2 ss2rabdv 
 |-  ( A C_ B -> { x e. SH | B C_ x } C_ { x e. SH | A C_ x } )
4 intss 
 |-  ( { x e. SH | B C_ x } C_ { x e. SH | A C_ x } -> |^| { x e. SH | A C_ x } C_ |^| { x e. SH | B C_ x } )
5 3 4 syl 
 |-  ( A C_ B -> |^| { x e. SH | A C_ x } C_ |^| { x e. SH | B C_ x } )
6 5 adantl 
 |-  ( ( B C_ ~H /\ A C_ B ) -> |^| { x e. SH | A C_ x } C_ |^| { x e. SH | B C_ x } )
7 sstr 
 |-  ( ( A C_ B /\ B C_ ~H ) -> A C_ ~H )
8 7 ancoms 
 |-  ( ( B C_ ~H /\ A C_ B ) -> A C_ ~H )
9 spanval 
 |-  ( A C_ ~H -> ( span ` A ) = |^| { x e. SH | A C_ x } )
10 8 9 syl 
 |-  ( ( B C_ ~H /\ A C_ B ) -> ( span ` A ) = |^| { x e. SH | A C_ x } )
11 spanval 
 |-  ( B C_ ~H -> ( span ` B ) = |^| { x e. SH | B C_ x } )
12 11 adantr 
 |-  ( ( B C_ ~H /\ A C_ B ) -> ( span ` B ) = |^| { x e. SH | B C_ x } )
13 6 10 12 3sstr4d 
 |-  ( ( B C_ ~H /\ A C_ B ) -> ( span ` A ) C_ ( span ` B ) )