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Theorem cmbr 28744
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 2819 . . . . 5 (𝑥 = 𝐴 → (𝑥C𝐴C ))
21anbi1d 743 . . . 4 (𝑥 = 𝐴 → ((𝑥C𝑦C ) ↔ (𝐴C𝑦C )))
3 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
4 ineq1 3942 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
5 ineq1 3942 . . . . . 6 (𝑥 = 𝐴 → (𝑥 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝑦)))
64, 5oveq12d 6823 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))) = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))))
73, 6eqeq12d 2767 . . . 4 (𝑥 = 𝐴 → (𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦)))))
82, 7anbi12d 749 . . 3 (𝑥 = 𝐴 → (((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦)))) ↔ ((𝐴C𝑦C ) ∧ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))))))
9 eleq1 2819 . . . . 5 (𝑦 = 𝐵 → (𝑦C𝐵C ))
109anbi2d 742 . . . 4 (𝑦 = 𝐵 → ((𝐴C𝑦C ) ↔ (𝐴C𝐵C )))
11 ineq2 3943 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
12 fveq2 6344 . . . . . . 7 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1312ineq2d 3949 . . . . . 6 (𝑦 = 𝐵 → (𝐴 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝐵)))
1411, 13oveq12d 6823 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))
1514eqeq2d 2762 . . . 4 (𝑦 = 𝐵 → (𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
1610, 15anbi12d 749 . . 3 (𝑦 = 𝐵 → (((𝐴C𝑦C ) ∧ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦)))) ↔ ((𝐴C𝐵C ) ∧ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))))
17 df-cm 28743 . . 3 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
188, 16, 17brabg 5136 . 2 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵 ↔ ((𝐴C𝐵C ) ∧ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))))
1918bianabs 960 1 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  cin 3706   class class class wbr 4796  cfv 6041  (class class class)co 6805   C cch 28087  cort 28088   chj 28091   𝐶 ccm 28094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-iota 6004  df-fv 6049  df-ov 6808  df-cm 28743
This theorem is referenced by:  cmbri  28750  cm2j  28780
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