Theorem cmbri 31568

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	⊢ 𝐴 ∈ Cℋ
pjoml2.2	⊢ 𝐵 ∈ Cℋ

Assertion

Ref	Expression
cmbri	⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))

Proof of Theorem cmbri

Step	Hyp	Ref	Expression
1		pjoml2.1	. 2 ⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	. 2 ⊢ 𝐵 ∈ Cℋ
3		cmbr 31562	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111   ∩ cin 3901   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346   C_ℋ cch 30907  ⊥cort 30908   ∨_ℋ chj 30911   𝐶_ℋ ccm 30914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-iota 6437  df-fv 6489  df-ov 7349  df-cm 31561

This theorem is referenced by:  cmcmlem  31569  cmcm2i  31571  cmbr2i  31574  cmbr3i  31578  pjclem1  32173  pjci  32178