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Theorem cmbri 29004
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 28998 . 2 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
41, 2, 3mp2an 685 1 (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1658  wcel 2166  cin 3797   class class class wbr 4873  cfv 6123  (class class class)co 6905   C cch 28341  cort 28342   chj 28345   𝐶 ccm 28348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-iota 6086  df-fv 6131  df-ov 6908  df-cm 28997
This theorem is referenced by:  cmcmlem  29005  cmcm2i  29007  cmbr2i  29010  cmbr3i  29014  pjclem1  29609  pjci  29614
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