Theorem cmbr 5499

Description: Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20.

Hypotheses

Ref	Expression
pjoml2.1	|- A e. CH
pjoml2.2	|- B e. CH

Assertion

Ref	Expression
cmbr	|- (A Com B <-> A = ((A i^i B) vH (A i^i (_|_` B))))

Proof of Theorem cmbr

Step	Hyp	Ref	Expression
1		pjoml2.1	. 2 |- A e. CH
2		pjoml2.2	. 2 |- B e. CH
3		cmbrt 5494	. 2 |- ((A e. CH /\ B e. CH) -> (A Com B <-> A = ((A i^i B) vH (A i^i (_|_` B)))))
4	1, 2, 3	mp2an 520	1 |- (A Com B <-> A = ((A i^i B) vH (A i^i (_|_` B))))

Syntax hints:   <-> wb 127   = wceq 1091   e. wcel 1092   i^i cin 1486   class class class wbr 2054  ` cfv 2422  (class class class)co 3001  CHcch 4968  _|_cort 4969   vH chj 4972   Com ccm 4975

This theorem is referenced by:  cmcmlem 5500  cmcm2 5502  cmbr2 5505  cmbr3 5509  pjclem1 5649  pjc 5654

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-cm 5493

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