Theorem cmbri 29367

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 29361	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))

Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114   ∩ cin 3935   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156   C_ℋ cch 28706  ⊥cort 28707   ∨_ℋ chj 28710   𝐶_ℋ ccm 28713

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-iota 6314  df-fv 6363  df-ov 7159  df-cm 29360

This theorem is referenced by:  cmcmlem  29368  cmcm2i  29370  cmbr2i  29373  cmbr3i  29377  pjclem1  29972  pjci  29977