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Metamath Proof Explorer

Theorem shsubcl

Description: Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion shsubcl 
|- ( ( H e. SH /\ A e. H /\ B e. H ) -> ( A -h B ) e. H )

Proof

Step Hyp Ref Expression
1 shss 
 |-  ( H e. SH -> H C_ ~H )
2 1 sseld 
 |-  ( H e. SH -> ( A e. H -> A e. ~H ) )
3 1 sseld 
 |-  ( H e. SH -> ( B e. H -> B e. ~H ) )
4 2 3 anim12d 
 |-  ( H e. SH -> ( ( A e. H /\ B e. H ) -> ( A e. ~H /\ B e. ~H ) ) )
5 4 3impib 
 |-  ( ( H e. SH /\ A e. H /\ B e. H ) -> ( A e. ~H /\ B e. ~H ) )
6 hvsubval 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( A -h B ) = ( A +h ( -u 1 .h B ) ) )
7 5 6 syl 
 |-  ( ( H e. SH /\ A e. H /\ B e. H ) -> ( A -h B ) = ( A +h ( -u 1 .h B ) ) )
8 neg1cn 
 |-  -u 1 e. CC
9 shmulcl 
 |-  ( ( H e. SH /\ -u 1 e. CC /\ B e. H ) -> ( -u 1 .h B ) e. H )
10 8 9 mp3an2 
 |-  ( ( H e. SH /\ B e. H ) -> ( -u 1 .h B ) e. H )
11 10 3adant2 
 |-  ( ( H e. SH /\ A e. H /\ B e. H ) -> ( -u 1 .h B ) e. H )
12 shaddcl 
 |-  ( ( H e. SH /\ A e. H /\ ( -u 1 .h B ) e. H ) -> ( A +h ( -u 1 .h B ) ) e. H )
13 11 12 syld3an3 
 |-  ( ( H e. SH /\ A e. H /\ B e. H ) -> ( A +h ( -u 1 .h B ) ) e. H )
14 7 13 eqeltrd 
 |-  ( ( H e. SH /\ A e. H /\ B e. H ) -> ( A -h B ) e. H )