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Metamath Proof Explorer

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 )