Metamath Proof Explorer

Theorem dmdbr7ati

Description: Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005) (New usage is discouraged.)

Ref Expression
Hypotheses sumdmdi.1 
|- A e. CH
sumdmdi.2 
|- B e. CH
Assertion dmdbr7ati 
|- ( A MH* B <-> A. x e. HAtoms ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) )

Proof

Step Hyp Ref Expression
1 sumdmdi.1 
 |-  A e. CH
2 sumdmdi.2 
 |-  B e. CH
3 1 2 dmdbr6ati 
 |-  ( A MH* B <-> A. x e. HAtoms ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) )
4 inss1 
 |-  ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B )
5 sseq1 
 |-  ( ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) -> ( ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) <-> ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) ) )
6 4 5 mpbiri 
 |-  ( ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) -> ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) )
7 6 ralimi 
 |-  ( A. x e. HAtoms ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) -> A. x e. HAtoms ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) )
8 3 7 sylbi 
 |-  ( A MH* B -> A. x e. HAtoms ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) )
9 sseqin2 
 |-  ( x C_ ( A vH B ) <-> ( ( A vH B ) i^i x ) = x )
10 9 biimpi 
 |-  ( x C_ ( A vH B ) -> ( ( A vH B ) i^i x ) = x )
11 10 sseq1d 
 |-  ( x C_ ( A vH B ) -> ( ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) <-> x C_ ( ( ( x vH B ) i^i A ) vH B ) ) )
12 11 biimpcd 
 |-  ( ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) -> ( x C_ ( A vH B ) -> x C_ ( ( ( x vH B ) i^i A ) vH B ) ) )
13 12 ralimi 
 |-  ( A. x e. HAtoms ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) -> A. x e. HAtoms ( x C_ ( A vH B ) -> x C_ ( ( ( x vH B ) i^i A ) vH B ) ) )
14 1 2 dmdbr5ati 
 |-  ( A MH* B <-> A. x e. HAtoms ( x C_ ( A vH B ) -> x C_ ( ( ( x vH B ) i^i A ) vH B ) ) )
15 13 14 sylibr 
 |-  ( A. x e. HAtoms ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) -> A MH* B )
16 8 15 impbii 
 |-  ( A MH* B <-> A. x e. HAtoms ( ( A vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B ) )