Theorem cmcm2i

Description: Commutation with orthocomplement. Theorem 2.3(i) of Beran p. 39. (Contributed by NM, 4-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	cmcm2i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chincli	⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ
4	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
5	1 4	chincli	⊢ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ
6	3 5	chjcomi	⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ∨ℋ ( 𝐴 ∩ 𝐵 ) )
7	2	pjococi	⊢ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵
8	7	ineq2i	⊢ ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∩ 𝐵 )
9	8	oveq2i	⊢ ( ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) ) = ( ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ∨ℋ ( 𝐴 ∩ 𝐵 ) )
10	6 9	eqtr4i	⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
11	10	eqeq2i	⊢ ( 𝐴 = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) ↔ 𝐴 = ( ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) ) )
12	1 2	cmbri	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) )
13	1 4	cmbri	⊢ ( 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) ↔ 𝐴 = ( ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) ) )
14	11 12 13	3bitr4i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) )