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Metamath Proof Explorer

Theorem cmm1i

Description: A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

Ref Expression
Hypotheses pjoml2.1 
|- A e. CH
pjoml2.2 
|- B e. CH
Assertion cmm1i 
|- A C_H ( A i^i B )

Proof

Step Hyp Ref Expression
1 pjoml2.1 
 |-  A e. CH
2 pjoml2.2 
 |-  B e. CH
3 1 2 chincli 
 |-  ( A i^i B ) e. CH
4 inss1 
 |-  ( A i^i B ) C_ A
5 3 1 4 lecmii 
 |-  ( A i^i B ) C_H A
6 3 1 5 cmcmii 
 |-  A C_H ( A i^i B )