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Theorem cmbri 22171
Description: Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
Hypotheses
Ref Expression
pjoml2.1  |-  A  e. 
CH
pjoml2.2  |-  B  e. 
CH
Assertion
Ref Expression
cmbri  |-  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )

Proof of Theorem cmbri
StepHypRef Expression
1 pjoml2.1 . 2  |-  A  e. 
CH
2 pjoml2.2 . 2  |-  B  e. 
CH
3 cmbr 22165 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B ) ) ) ) )
41, 2, 3mp2an 653 1  |-  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1625    e. wcel 1686    i^i cin 3153   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CHcch 21511   _|_cort 21512    vH chj 21515    C_H ccm 21518
This theorem is referenced by:  cmcmlem  22172  cmcm2i  22174  cmbr2i  22177  cmbr3i  22181  pjclem1  22777  pjci  22782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-iota 5221  df-fv 5265  df-ov 5863  df-cm 22164
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