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Metamath Proof Explorer

Theorem cmbri

Description: Binary relation expressing the commutes relation. Definition of commutes in Kalmbach p. 20. (Contributed by NM, 6-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ C_ℋ
	pjoml2.2	⊢ 𝐵 ∈ C_ℋ
Assertion	cmbri	⊢ ( 𝐴 𝐶_ℋ 𝐵 ↔ 𝐴 = ( ( 𝐴 ∩ 𝐵 ) ∨_ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ C_ℋ
2		pjoml2.2	⊢ 𝐵 ∈ C_ℋ
3		cmbr	⊢ ( ( 𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ( 𝐴 𝐶_ℋ 𝐵 ↔ 𝐴 = ( ( 𝐴 ∩ 𝐵 ) ∨_ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) ) )
4	1 2 3	mp2an	⊢ ( 𝐴 𝐶_ℋ 𝐵 ↔ 𝐴 = ( ( 𝐴 ∩ 𝐵 ) ∨_ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) )