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Metamath Proof Explorer

Theorem cvati

Description: If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004) (New usage is discouraged.)

Ref Expression
Hypotheses chpssat.1 
|- A e. CH
chpssat.2 
|- B e. CH
Assertion cvati 
|- ( A  E. x e. HAtoms ( A vH x ) = B )

Proof

Step Hyp Ref Expression
1 chpssat.1 
 |-  A e. CH
2 chpssat.2 
 |-  B e. CH
3 cvpss 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A  A C. B ) )
4 1 2 3 mp2an 
 |-  ( A  A C. B )
5 1 2 chrelati 
 |-  ( A C. B -> E. x e. HAtoms ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) )
6 4 5 syl 
 |-  ( A  E. x e. HAtoms ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) )
7 cvnbtwn2 
 |-  ( ( A e. CH /\ B e. CH /\ ( A vH x ) e. CH ) -> ( A  ( ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) -> ( A vH x ) = B ) ) )
8 1 2 7 mp3an12 
 |-  ( ( A vH x ) e. CH -> ( A  ( ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) -> ( A vH x ) = B ) ) )
9 atelch 
 |-  ( x e. HAtoms -> x e. CH )
10 chjcl 
 |-  ( ( A e. CH /\ x e. CH ) -> ( A vH x ) e. CH )
11 1 9 10 sylancr 
 |-  ( x e. HAtoms -> ( A vH x ) e. CH )
12 8 11 syl11 
 |-  ( A  ( x e. HAtoms -> ( ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) -> ( A vH x ) = B ) ) )
13 12 reximdvai 
 |-  ( A  ( E. x e. HAtoms ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) -> E. x e. HAtoms ( A vH x ) = B ) )
14 6 13 mpd 
 |-  ( A  E. x e. HAtoms ( A vH x ) = B )