Metamath Proof Explorer


Theorem ch1dle

Description: A 1-dimensional subspace is less than or equal to any member of CH containing its generating vector. (Contributed by NM, 30-May-2004) (New usage is discouraged.)

Ref Expression
Assertion ch1dle
|- ( ( A e. CH /\ B e. A ) -> ( _|_ ` ( _|_ ` { B } ) ) C_ A )

Proof

Step Hyp Ref Expression
1 chsh
 |-  ( A e. CH -> A e. SH )
2 sh1dle
 |-  ( ( A e. SH /\ B e. A ) -> ( _|_ ` ( _|_ ` { B } ) ) C_ A )
3 1 2 sylan
 |-  ( ( A e. CH /\ B e. A ) -> ( _|_ ` ( _|_ ` { B } ) ) C_ A )