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Theorem cmbr 31613
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 2827 . . . . 5 (𝑥 = 𝐴 → (𝑥C𝐴C ))
21anbi1d 631 . . . 4 (𝑥 = 𝐴 → ((𝑥C𝑦C ) ↔ (𝐴C𝑦C )))
3 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
4 ineq1 4221 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
5 ineq1 4221 . . . . . 6 (𝑥 = 𝐴 → (𝑥 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝑦)))
64, 5oveq12d 7449 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))) = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))))
73, 6eqeq12d 2751 . . . 4 (𝑥 = 𝐴 → (𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦)))))
82, 7anbi12d 632 . . 3 (𝑥 = 𝐴 → (((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦)))) ↔ ((𝐴C𝑦C ) ∧ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))))))
9 eleq1 2827 . . . . 5 (𝑦 = 𝐵 → (𝑦C𝐵C ))
109anbi2d 630 . . . 4 (𝑦 = 𝐵 → ((𝐴C𝑦C ) ↔ (𝐴C𝐵C )))
11 ineq2 4222 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
12 fveq2 6907 . . . . . . 7 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1312ineq2d 4228 . . . . . 6 (𝑦 = 𝐵 → (𝐴 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝐵)))
1411, 13oveq12d 7449 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))
1514eqeq2d 2746 . . . 4 (𝑦 = 𝐵 → (𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
1610, 15anbi12d 632 . . 3 (𝑦 = 𝐵 → (((𝐴C𝑦C ) ∧ 𝐴 = ((𝐴𝑦) ∨ (𝐴 ∩ (⊥‘𝑦)))) ↔ ((𝐴C𝐵C ) ∧ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))))
17 df-cm 31612 . . 3 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
188, 16, 17brabg 5549 . 2 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵 ↔ ((𝐴C𝐵C ) ∧ 𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵))))))
1918bianabs 541 1 ((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962   class class class wbr 5148  cfv 6563  (class class class)co 7431   C cch 30958  cort 30959   chj 30962   𝐶 ccm 30965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434  df-cm 31612
This theorem is referenced by:  cmbri  31619  cm2j  31649
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