Theorem cmbr 9490

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 C_H 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 9484	. 2 |- ((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 696	1 |- (A C_H B <-> A = ((A i^i B) vH (A i^i (_|_` B))))

Syntax hints:   <-> wb 146   = wceq 955   e. wcel 957   i^i cin 2043   class class class wbr 2615  ` cfv 3178  (class class class)co 3958  CHcch 8753  _|_cort 8754   vH chj 8757   C_H ccm 8760

This theorem is referenced by:  cmcmlem 9491  cmcm2 9493  cmbr2 9496  cmbr3 9500  pjclem1 10079  pjc 10084

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194  df-opr 3960  df-cm 9483

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