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Metamath Proof Explorer

Theorem chabs1

Description: Hilbert lattice absorption law. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 15-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chabs1 
|- ( ( A e. CH /\ B e. CH ) -> ( A vH ( A i^i B ) ) = A )

Proof

Step Hyp Ref Expression
1 ssid 
 |-  A C_ A
2 inss1 
 |-  ( A i^i B ) C_ A
3 1 2 pm3.2i 
 |-  ( A C_ A /\ ( A i^i B ) C_ A )
4 simpl 
 |-  ( ( A e. CH /\ B e. CH ) -> A e. CH )
5 chincl 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A i^i B ) e. CH )
6 chlub 
 |-  ( ( A e. CH /\ ( A i^i B ) e. CH /\ A e. CH ) -> ( ( A C_ A /\ ( A i^i B ) C_ A ) <-> ( A vH ( A i^i B ) ) C_ A ) )
7 4 5 4 6 syl3anc 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( A C_ A /\ ( A i^i B ) C_ A ) <-> ( A vH ( A i^i B ) ) C_ A ) )
8 3 7 mpbii 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A vH ( A i^i B ) ) C_ A )
9 chub1 
 |-  ( ( A e. CH /\ ( A i^i B ) e. CH ) -> A C_ ( A vH ( A i^i B ) ) )
10 5 9 syldan 
 |-  ( ( A e. CH /\ B e. CH ) -> A C_ ( A vH ( A i^i B ) ) )
11 8 10 eqssd 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A vH ( A i^i B ) ) = A )