Theorem mddmd

Description: The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	mddmd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
2		choccl	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
3		dmdmd	⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) 𝑀ℋ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) 𝑀ℋ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
5		ococ	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )
6		ococ	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
7	5 6	breqan12d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) 𝑀ℋ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ↔ 𝐴 𝑀ℋ 𝐵 ) )
8	4 7	bitr2d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) ) )