Theorem cmbr 9450

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 9444	. 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 695	1 |- (A C_H B <-> A = ((A i^i B) vH (A i^i (_|_` B))))

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955   i^i cin 2036   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CHcch 8737  _|_cort 8738   vH chj 8741   C_H ccm 8744

This theorem is referenced by:  cmcmlem 9451  cmcm2 9453  cmbr2 9456  cmbr3 9460  pjclem1 10033  pjc 10038

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950  df-cm 9443

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