Theorem cmbri 32197

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 32191	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144   ∩ cin 4089   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420   C_ℋ cch 31536  ⊥cort 31537   ∨_ℋ chj 31540   𝐶_ℋ ccm 31543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6494  df-fv 6546  df-ov 7423  df-cm 32190

This theorem is used by:  cmcmlem  32198  cmcm2i  32200  cmbr2i  32203  cmbr3i  32207  pjclem1  32802  pjci  32807