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 } ) ) )