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Metamath Proof Explorer

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 ) ) )