Metamath Proof Explorer

Theorem dmdbr6ati

Description: Dual modular pair property in terms of atoms. The modular law takes the form of the shearing identity. (Contributed by NM, 18-Jan-2005) (New usage is discouraged.)

Ref Expression
Hypotheses sumdmdi.1 
|- A e. CH
sumdmdi.2 
|- B e. CH
Assertion 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 ) )

Proof

Step Hyp Ref Expression
1 sumdmdi.1 
 |-  A e. CH
2 sumdmdi.2 
 |-  B e. CH
3 dmdbr3 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A MH* B <-> A. x e. CH ( ( ( x vH B ) i^i A ) vH B ) = ( ( x vH B ) i^i ( A vH B ) ) ) )
4 1 2 3 mp2an 
 |-  ( A MH* B <-> A. x e. CH ( ( ( x vH B ) i^i A ) vH B ) = ( ( x vH B ) i^i ( A vH B ) ) )
5 chabs2 
 |-  ( ( x e. CH /\ B e. CH ) -> ( x i^i ( x vH B ) ) = x )
6 2 5 mpan2 
 |-  ( x e. CH -> ( x i^i ( x vH B ) ) = x )
7 6 ineq2d 
 |-  ( x e. CH -> ( ( A vH B ) i^i ( x i^i ( x vH B ) ) ) = ( ( A vH B ) i^i x ) )
8 incom 
 |-  ( ( A vH B ) i^i ( x i^i ( x vH B ) ) ) = ( ( x i^i ( x vH B ) ) i^i ( A vH B ) )
9 inass 
 |-  ( ( x i^i ( x vH B ) ) i^i ( A vH B ) ) = ( x i^i ( ( x vH B ) i^i ( A vH B ) ) )
10 incom 
 |-  ( x i^i ( ( x vH B ) i^i ( A vH B ) ) ) = ( ( ( x vH B ) i^i ( A vH B ) ) i^i x )
11 8 9 10 3eqtri 
 |-  ( ( A vH B ) i^i ( x i^i ( x vH B ) ) ) = ( ( ( x vH B ) i^i ( A vH B ) ) i^i x )
12 7 11 eqtr3di 
 |-  ( x e. CH -> ( ( A vH B ) i^i x ) = ( ( ( x vH B ) i^i ( A vH B ) ) i^i x ) )
13 12 adantr 
 |-  ( ( x e. CH /\ ( ( ( x vH B ) i^i A ) vH B ) = ( ( x vH B ) i^i ( A vH B ) ) ) -> ( ( A vH B ) i^i x ) = ( ( ( x vH B ) i^i ( A vH B ) ) i^i x ) )
14 ineq1 
 |-  ( ( ( ( x vH B ) i^i A ) vH B ) = ( ( x vH B ) i^i ( A vH B ) ) -> ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) = ( ( ( x vH B ) i^i ( A vH B ) ) i^i x ) )
15 14 adantl 
 |-  ( ( x e. CH /\ ( ( ( x vH B ) i^i A ) vH B ) = ( ( x vH B ) i^i ( A vH B ) ) ) -> ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) = ( ( ( x vH B ) i^i ( A vH B ) ) i^i x ) )
16 13 15 eqtr4d 
 |-  ( ( x e. CH /\ ( ( ( x vH B ) i^i A ) vH B ) = ( ( x vH B ) i^i ( A vH B ) ) ) -> ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) )
17 16 ralimiaa 
 |-  ( A. x e. CH ( ( ( x vH B ) i^i A ) vH B ) = ( ( x vH B ) i^i ( A vH B ) ) -> A. x e. CH ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) )
18 4 17 sylbi 
 |-  ( A MH* B -> A. x e. CH ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) )
19 atelch 
 |-  ( x e. HAtoms -> x e. CH )
20 19 imim1i 
 |-  ( ( x e. CH -> ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) ) -> ( x e. HAtoms -> ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) ) )
21 20 ralimi2 
 |-  ( A. x e. CH ( ( 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 ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) )
22 18 21 syl 
 |-  ( A MH* B -> A. x e. HAtoms ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) )
23 inss1 
 |-  ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) C_ ( ( ( x vH B ) i^i A ) vH B )
24 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 ) ) )
25 23 24 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 ) )
26 incom 
 |-  ( ( A vH B ) i^i x ) = ( x i^i ( A vH B ) )
27 df-ss 
 |-  ( x C_ ( A vH B ) <-> ( x i^i ( A vH B ) ) = x )
28 27 biimpi 
 |-  ( x C_ ( A vH B ) -> ( x i^i ( A vH B ) ) = x )
29 26 28 eqtrid 
 |-  ( x C_ ( A vH B ) -> ( ( A vH B ) i^i x ) = x )
30 29 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 ) ) )
31 25 30 syl5ibcom 
 |-  ( ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) -> ( x C_ ( A vH B ) -> x C_ ( ( ( x vH B ) i^i A ) vH B ) ) )
32 31 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 ( x C_ ( A vH B ) -> x C_ ( ( ( x vH B ) i^i A ) vH B ) ) )
33 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 ) ) )
34 32 33 sylibr 
 |-  ( A. x e. HAtoms ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) -> A MH* B )
35 22 34 impbii 
 |-  ( A MH* B <-> A. x e. HAtoms ( ( A vH B ) i^i x ) = ( ( ( ( x vH B ) i^i A ) vH B ) i^i x ) )