This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem shsel1

Description: A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

Ref Expression
Assertion shsel1 
|- ( ( A e. SH /\ B e. SH ) -> ( C e. A -> C e. ( A +H B ) ) )

Proof

Step Hyp Ref Expression
1 shel 
 |-  ( ( A e. SH /\ C e. A ) -> C e. ~H )
2 ax-hvaddid 
 |-  ( C e. ~H -> ( C +h 0h ) = C )
3 1 2 syl 
 |-  ( ( A e. SH /\ C e. A ) -> ( C +h 0h ) = C )
4 3 adantlr 
 |-  ( ( ( A e. SH /\ B e. SH ) /\ C e. A ) -> ( C +h 0h ) = C )
5 sh0 
 |-  ( B e. SH -> 0h e. B )
6 5 adantl 
 |-  ( ( A e. SH /\ B e. SH ) -> 0h e. B )
7 shsva 
 |-  ( ( A e. SH /\ B e. SH ) -> ( ( C e. A /\ 0h e. B ) -> ( C +h 0h ) e. ( A +H B ) ) )
8 6 7 mpan2d 
 |-  ( ( A e. SH /\ B e. SH ) -> ( C e. A -> ( C +h 0h ) e. ( A +H B ) ) )
9 8 imp 
 |-  ( ( ( A e. SH /\ B e. SH ) /\ C e. A ) -> ( C +h 0h ) e. ( A +H B ) )
10 4 9 eqeltrrd 
 |-  ( ( ( A e. SH /\ B e. SH ) /\ C e. A ) -> C e. ( A +H B ) )
11 10 ex 
 |-  ( ( A e. SH /\ B e. SH ) -> ( C e. A -> C e. ( A +H B ) ) )