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Theorem cmbr 28289
Description: Binary relation expressing 𝐴 commutes with 𝐵. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cmbr ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))

Proof of Theorem cmbr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2686 . . . . 5 (𝑥 = 𝐴 → (𝑥C𝐴C ))
21anbi1d 740 . . . 4 (𝑥 = 𝐴 → ((𝑥C𝑦C ) ↔ (𝐴C𝑦C )))
3 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
4 ineq1 3785 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
5 ineq1 3785 . . . . . 6 (𝑥 = 𝐴 → (𝑥 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝑦)))
64, 5oveq12d 6622 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))) = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))))
73, 6eqeq12d 2636 . . . 4 (𝑥 = 𝐴 → (𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦)))))
82, 7anbi12d 746 . . 3 (𝑥 = 𝐴 → (((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦)))) ↔ ((𝐴C𝑦C ) ∧ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))))))
9 eleq1 2686 . . . . 5 (𝑦 = 𝐵 → (𝑦C𝐵C ))
109anbi2d 739 . . . 4 (𝑦 = 𝐵 → ((𝐴C𝑦C ) ↔ (𝐴C𝐵C )))
11 ineq2 3786 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
12 fveq2 6148 . . . . . . 7 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1312ineq2d 3792 . . . . . 6 (𝑦 = 𝐵 → (𝐴 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝐵)))
1411, 13oveq12d 6622 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))
1514eqeq2d 2631 . . . 4 (𝑦 = 𝐵 → (𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
1610, 15anbi12d 746 . . 3 (𝑦 = 𝐵 → (((𝐴C𝑦C ) ∧ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦)))) ↔ ((𝐴C𝐵C ) ∧ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))))
17 df-cm 28288 . . 3 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
188, 16, 17brabg 4954 . 2 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵 ↔ ((𝐴C𝐵C ) ∧ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))))
1918bianabs 923 1 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cin 3554   class class class wbr 4613  cfv 5847  (class class class)co 6604   C cch 27632  cort 27633   chj 27636   𝐶 ccm 27639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-iota 5810  df-fv 5855  df-ov 6607  df-cm 28288
This theorem is referenced by:  cmbri  28295  cm2j  28325
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