Metamath Proof Explorer


Theorem cmbri

Description: Binary relation expressing the commutes relation. Definition of commutes in Kalmbach p. 20. (Contributed by NM, 6-Aug-2004) (New usage is discouraged.)

Ref Expression
Hypotheses pjoml2.1
|- A e. CH
pjoml2.2
|- B e. CH
Assertion cmbri
|- ( A C_H B <-> A = ( ( A i^i B ) vH ( A i^i ( _|_ ` B ) ) ) )

Proof

Step Hyp Ref Expression
1 pjoml2.1
 |-  A e. CH
2 pjoml2.2
 |-  B e. CH
3 cmbr
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> A = ( ( A i^i B ) vH ( A i^i ( _|_ ` B ) ) ) ) )
4 1 2 3 mp2an
 |-  ( A C_H B <-> A = ( ( A i^i B ) vH ( A i^i ( _|_ ` B ) ) ) )