Theorem mdsym

Description: M-symmetry of the Hilbert lattice. Lemma 5 of Maeda p. 168. (Contributed by NM, 6-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	mdsym	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ) )
2		breq2	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐵 𝑀ℋ 𝐴 ↔ 𝐵 𝑀ℋ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
3	1 2	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) )
4		breq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
5		breq1	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( 𝐵 𝑀ℋ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) 𝑀ℋ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
6	4 5	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) 𝑀ℋ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) )
7		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
8		ifchhv	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
9	7 8	mdsymi	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) 𝑀ℋ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) )
10	3 6 9	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴 ) )