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Theorem cmbri 31609
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 31603 . 2 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
41, 2, 3mp2an 692 1 (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  cin 3950   class class class wbr 5143  cfv 6561  (class class class)co 7431   C cch 30948  cort 30949   chj 30952   𝐶 ccm 30955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434  df-cm 31602
This theorem is referenced by:  cmcmlem  31610  cmcm2i  31612  cmbr2i  31615  cmbr3i  31619  pjclem1  32214  pjci  32219
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