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Metamath Proof Explorer

Theorem spansnj

Description: The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Contributed by NM, 4-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion spansnj 
|- ( ( A e. CH /\ B e. ~H ) -> ( A +H ( span ` { B } ) ) = ( A vH ( span ` { B } ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A +H ( span ` { B } ) ) = ( if ( A e. CH , A , ~H ) +H ( span ` { B } ) ) )
2 oveq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A vH ( span ` { B } ) ) = ( if ( A e. CH , A , ~H ) vH ( span ` { B } ) ) )
3 1 2 eqeq12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A +H ( span ` { B } ) ) = ( A vH ( span ` { B } ) ) <-> ( if ( A e. CH , A , ~H ) +H ( span ` { B } ) ) = ( if ( A e. CH , A , ~H ) vH ( span ` { B } ) ) ) )
4 sneq 
 |-  ( B = if ( B e. ~H , B , 0h ) -> { B } = { if ( B e. ~H , B , 0h ) } )
5 4 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( span ` { B } ) = ( span ` { if ( B e. ~H , B , 0h ) } ) )
6 5 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. CH , A , ~H ) +H ( span ` { B } ) ) = ( if ( A e. CH , A , ~H ) +H ( span ` { if ( B e. ~H , B , 0h ) } ) ) )
7 5 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. CH , A , ~H ) vH ( span ` { B } ) ) = ( if ( A e. CH , A , ~H ) vH ( span ` { if ( B e. ~H , B , 0h ) } ) ) )
8 6 7 eqeq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( if ( A e. CH , A , ~H ) +H ( span ` { B } ) ) = ( if ( A e. CH , A , ~H ) vH ( span ` { B } ) ) <-> ( if ( A e. CH , A , ~H ) +H ( span ` { if ( B e. ~H , B , 0h ) } ) ) = ( if ( A e. CH , A , ~H ) vH ( span ` { if ( B e. ~H , B , 0h ) } ) ) ) )
9 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
10 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
11 9 10 spansnji 
 |-  ( if ( A e. CH , A , ~H ) +H ( span ` { if ( B e. ~H , B , 0h ) } ) ) = ( if ( A e. CH , A , ~H ) vH ( span ` { if ( B e. ~H , B , 0h ) } ) )
12 3 8 11 dedth2h 
 |-  ( ( A e. CH /\ B e. ~H ) -> ( A +H ( span ` { B } ) ) = ( A vH ( span ` { B } ) ) )