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Metamath Proof Explorer

Theorem sspadd1

Description: A projective subspace sum is a superset of its first summand. ( ssun1 analog.) (Contributed by NM, 3-Jan-2012)

Ref Expression
Hypotheses padd0.a 
|- A = ( Atoms ` K )
padd0.p 
|- .+ = ( +P ` K )
Assertion sspadd1 
|- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) )

Proof

Step Hyp Ref Expression
1 padd0.a 
 |-  A = ( Atoms ` K )
2 padd0.p 
 |-  .+ = ( +P ` K )
3 ssun1 
 |-  X C_ ( X u. Y )
4 ssun1 
 |-  ( X u. Y ) C_ ( ( X u. Y ) u. { p e. A | E. q e. X E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) } )
5 3 4 sstri 
 |-  X C_ ( ( X u. Y ) u. { p e. A | E. q e. X E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) } )
6 eqid 
 |-  ( le ` K ) = ( le ` K )
7 eqid 
 |-  ( join ` K ) = ( join ` K )
8 6 7 1 2 paddval 
 |-  ( ( K e. B /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) = ( ( X u. Y ) u. { p e. A | E. q e. X E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) } ) )
9 5 8 sseqtrrid 
 |-  ( ( K e. B /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) )