Metamath Proof Explorer

Theorem shslej

Description: Subspace sum is smaller than subspace join. Remark in Kalmbach p. 65. (Contributed by NM, 12-Jul-2004) (New usage is discouraged.)

Ref Expression
Assertion shslej 
|- ( ( A e. SH /\ B e. SH ) -> ( A +H B ) C_ ( A vH B ) )

Proof

Step Hyp Ref Expression
1 oveq1 
 |-  ( A = if ( A e. SH , A , ~H ) -> ( A +H B ) = ( if ( A e. SH , A , ~H ) +H B ) )
2 oveq1 
 |-  ( A = if ( A e. SH , A , ~H ) -> ( A vH B ) = ( if ( A e. SH , A , ~H ) vH B ) )
3 1 2 sseq12d 
 |-  ( A = if ( A e. SH , A , ~H ) -> ( ( A +H B ) C_ ( A vH B ) <-> ( if ( A e. SH , A , ~H ) +H B ) C_ ( if ( A e. SH , A , ~H ) vH B ) ) )
4 oveq2 
 |-  ( B = if ( B e. SH , B , ~H ) -> ( if ( A e. SH , A , ~H ) +H B ) = ( if ( A e. SH , A , ~H ) +H if ( B e. SH , B , ~H ) ) )
5 oveq2 
 |-  ( B = if ( B e. SH , B , ~H ) -> ( if ( A e. SH , A , ~H ) vH B ) = ( if ( A e. SH , A , ~H ) vH if ( B e. SH , B , ~H ) ) )
6 4 5 sseq12d 
 |-  ( B = if ( B e. SH , B , ~H ) -> ( ( if ( A e. SH , A , ~H ) +H B ) C_ ( if ( A e. SH , A , ~H ) vH B ) <-> ( if ( A e. SH , A , ~H ) +H if ( B e. SH , B , ~H ) ) C_ ( if ( A e. SH , A , ~H ) vH if ( B e. SH , B , ~H ) ) ) )
7 helsh 
 |-  ~H e. SH
8 7 elimel 
 |-  if ( A e. SH , A , ~H ) e. SH
9 7 elimel 
 |-  if ( B e. SH , B , ~H ) e. SH
10 8 9 shsleji 
 |-  ( if ( A e. SH , A , ~H ) +H if ( B e. SH , B , ~H ) ) C_ ( if ( A e. SH , A , ~H ) vH if ( B e. SH , B , ~H ) )
11 3 6 10 dedth2h 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A +H B ) C_ ( A vH B ) )