Theorem cmbri 31526

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 31520	. 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 1540   ∈ wcel 2109   ∩ cin 3916   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390   C_ℋ cch 30865  ⊥cort 30866   ∨_ℋ chj 30869   𝐶_ℋ ccm 30872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-iota 6467  df-fv 6522  df-ov 7393  df-cm 31519

This theorem is referenced by:  cmcmlem  31527  cmcm2i  31529  cmbr2i  31532  cmbr3i  31536  pjclem1  32131  pjci  32136