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Metamath Proof Explorer

Theorem chjatom

Description: The join of a closed subspace and an atom equals their subspace sum. Special case of remark in Kalmbach p. 65, stating that if A or B is finite-dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chjatom 
|- ( ( A e. CH /\ B e. HAtoms ) -> ( A +H B ) = ( A vH B ) )

Proof

Step Hyp Ref Expression
1 atom1d 
 |-  ( B e. HAtoms <-> E. x e. ~H ( x =/= 0h /\ B = ( span ` { x } ) ) )
2 spansnj 
 |-  ( ( A e. CH /\ x e. ~H ) -> ( A +H ( span ` { x } ) ) = ( A vH ( span ` { x } ) ) )
3 oveq2 
 |-  ( B = ( span ` { x } ) -> ( A +H B ) = ( A +H ( span ` { x } ) ) )
4 oveq2 
 |-  ( B = ( span ` { x } ) -> ( A vH B ) = ( A vH ( span ` { x } ) ) )
5 3 4 eqeq12d 
 |-  ( B = ( span ` { x } ) -> ( ( A +H B ) = ( A vH B ) <-> ( A +H ( span ` { x } ) ) = ( A vH ( span ` { x } ) ) ) )
6 2 5 imbitrrid 
 |-  ( B = ( span ` { x } ) -> ( ( A e. CH /\ x e. ~H ) -> ( A +H B ) = ( A vH B ) ) )
7 6 expd 
 |-  ( B = ( span ` { x } ) -> ( A e. CH -> ( x e. ~H -> ( A +H B ) = ( A vH B ) ) ) )
8 7 adantl 
 |-  ( ( x =/= 0h /\ B = ( span ` { x } ) ) -> ( A e. CH -> ( x e. ~H -> ( A +H B ) = ( A vH B ) ) ) )
9 8 com3l 
 |-  ( A e. CH -> ( x e. ~H -> ( ( x =/= 0h /\ B = ( span ` { x } ) ) -> ( A +H B ) = ( A vH B ) ) ) )
10 9 rexlimdv 
 |-  ( A e. CH -> ( E. x e. ~H ( x =/= 0h /\ B = ( span ` { x } ) ) -> ( A +H B ) = ( A vH B ) ) )
11 1 10 biimtrid 
 |-  ( A e. CH -> ( B e. HAtoms -> ( A +H B ) = ( A vH B ) ) )
12 11 imp 
 |-  ( ( A e. CH /\ B e. HAtoms ) -> ( A +H B ) = ( A vH B ) )