Metamath Proof Explorer

Theorem chrelati

Description: The Hilbert lattice is relatively atomic. Remark 2 of Kalmbach p. 149. (Contributed by NM, 11-Jun-2004) (New usage is discouraged.)

Ref Expression
Hypotheses chpssat.1 
|- A e. CH
chpssat.2 
|- B e. CH
Assertion chrelati 
|- ( A C. B -> E. x e. HAtoms ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) )

Proof

Step Hyp Ref Expression
1 chpssat.1 
 |-  A e. CH
2 chpssat.2 
 |-  B e. CH
3 1 2 chpssati 
 |-  ( A C. B -> E. x e. HAtoms ( x C_ B /\ -. x C_ A ) )
4 ancom 
 |-  ( ( x C_ B /\ -. x C_ A ) <-> ( -. x C_ A /\ x C_ B ) )
5 pssss 
 |-  ( A C. B -> A C_ B )
6 atelch 
 |-  ( x e. HAtoms -> x e. CH )
7 chnle 
 |-  ( ( A e. CH /\ x e. CH ) -> ( -. x C_ A <-> A C. ( A vH x ) ) )
8 1 7 mpan 
 |-  ( x e. CH -> ( -. x C_ A <-> A C. ( A vH x ) ) )
9 8 adantl 
 |-  ( ( A C_ B /\ x e. CH ) -> ( -. x C_ A <-> A C. ( A vH x ) ) )
10 ibar 
 |-  ( A C_ B -> ( x C_ B <-> ( A C_ B /\ x C_ B ) ) )
11 chlub 
 |-  ( ( A e. CH /\ x e. CH /\ B e. CH ) -> ( ( A C_ B /\ x C_ B ) <-> ( A vH x ) C_ B ) )
12 1 2 11 mp3an13 
 |-  ( x e. CH -> ( ( A C_ B /\ x C_ B ) <-> ( A vH x ) C_ B ) )
13 10 12 sylan9bb 
 |-  ( ( A C_ B /\ x e. CH ) -> ( x C_ B <-> ( A vH x ) C_ B ) )
14 9 13 anbi12d 
 |-  ( ( A C_ B /\ x e. CH ) -> ( ( -. x C_ A /\ x C_ B ) <-> ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) ) )
15 5 6 14 syl2an 
 |-  ( ( A C. B /\ x e. HAtoms ) -> ( ( -. x C_ A /\ x C_ B ) <-> ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) ) )
16 4 15 bitrid 
 |-  ( ( A C. B /\ x e. HAtoms ) -> ( ( x C_ B /\ -. x C_ A ) <-> ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) ) )
17 16 rexbidva 
 |-  ( A C. B -> ( E. x e. HAtoms ( x C_ B /\ -. x C_ A ) <-> E. x e. HAtoms ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) ) )
18 3 17 mpbid 
 |-  ( A C. B -> E. x e. HAtoms ( A C. ( A vH x ) /\ ( A vH x ) C_ B ) )