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Theorem cmbri 31312
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 𝐴C
pjoml2.2 𝐵C
Assertion
Ref Expression
cmbri (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))

Proof of Theorem cmbri
StepHypRef Expression
1 pjoml2.1 . 2 𝐴C
2 pjoml2.2 . 2 𝐵C
3 cmbr 31306 . 2 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
41, 2, 3mp2an 689 1 (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  cin 3939   class class class wbr 5138  cfv 6533  (class class class)co 7401   C cch 30651  cort 30652   chj 30655   𝐶 ccm 30658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-iota 6485  df-fv 6541  df-ov 7404  df-cm 31305
This theorem is referenced by:  cmcmlem  31313  cmcm2i  31315  cmbr2i  31318  cmbr3i  31322  pjclem1  31917  pjci  31922
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