Theorem cvdmd

Description: The covering property implies the dual modular pair property. Lemma 7.5.2 of MaedaMaeda p. 31. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvdmd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐵 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) → 𝐴 𝑀ℋ* 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
2		choccl	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
3		cvmd	⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ∧ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) )
4	3	3expia	⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
5	1 2 4	syl2an	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
6		simpr	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐵 ∈ Cℋ )
7		chjcl	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )
8		cvcon3	⊢ ( ( 𝐵 ∈ Cℋ ∧ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ ) → ( 𝐵 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ↔ ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) ) )
9	6 7 8	syl2anc	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ↔ ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) ) )
10		chdmj1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) )
11	10	breq1d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) ↔ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) ) )
12	9 11	bitrd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ↔ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ⋖ℋ ( ⊥ ‘ 𝐵 ) ) )
13		dmdmd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
14	5 12 13	3imtr4d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) → 𝐴 𝑀ℋ* 𝐵 ) )
15	14	3impia	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐵 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) → 𝐴 𝑀ℋ* 𝐵 )