Metamath Proof Explorer

Theorem spansn0

Description: The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005) (New usage is discouraged.)

Ref Expression
Assertion spansn0 
|- ( span ` { 0h } ) = 0H

Proof

Step Hyp Ref Expression
1 df-ch0 
 |-  0H = { 0h }
2 1 fveq2i 
 |-  ( span ` 0H ) = ( span ` { 0h } )
3 h0elsh 
 |-  0H e. SH
4 spanid 
 |-  ( 0H e. SH -> ( span ` 0H ) = 0H )
5 3 4 ax-mp 
 |-  ( span ` 0H ) = 0H
6 2 5 eqtr3i 
 |-  ( span ` { 0h } ) = 0H