Theorem cmcm3

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	cmcm3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 𝐶ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ) )
2		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) )
3	2	breq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( ⊥ ‘ 𝐴 ) 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) 𝐶ℋ 𝐵 ) )
4	1 3	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) 𝐶ℋ 𝐵 ) ) )
5		breq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
6		breq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
7	5 6	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) 𝐶ℋ 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ↔ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) )
8		h0elch	⊢ 0ℋ ∈ Cℋ
9	8	elimel	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∈ Cℋ
10	8	elimel	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ∈ Cℋ
11	9 10	cmcm3i	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ↔ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) )
12	4 7 11	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ 𝐵 ) )