# Metamath Proof Explorer

## Theorem mdsl3

Description: Sublattice mapping for a modular pair. Part of Theorem 1.3 of MaedaMaeda p. 2. (Contributed by NM, 26-Apr-2006) (New usage is discouraged.)

Ref Expression
Assertion mdsl3
`|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ ( A i^i B ) C_ C /\ C C_ B ) ) -> ( ( C vH A ) i^i B ) = C )`

### Proof

Step Hyp Ref Expression
1 mdi
` |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ C C_ B ) ) -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) )`
` |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ ( A i^i B ) C_ C /\ C C_ B ) ) -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) )`
` |-  ( ( A e. CH /\ B e. CH ) -> ( A i^i B ) e. CH )`
` |-  ( ( ( A i^i B ) e. CH /\ C e. CH ) -> ( ( A i^i B ) C_ C <-> ( C vH ( A i^i B ) ) = C ) )`
` |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A i^i B ) C_ C <-> ( C vH ( A i^i B ) ) = C ) )`
` |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A i^i B ) C_ C ) -> ( C vH ( A i^i B ) ) = C )`
` |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ ( A i^i B ) C_ C /\ C C_ B ) ) -> ( C vH ( A i^i B ) ) = C )`
` |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ ( A i^i B ) C_ C /\ C C_ B ) ) -> ( ( C vH A ) i^i B ) = C )`