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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 ∈ C ℋ ∧ B ∈ C ℋ ∧ B ⋖ ℋ A ∨ ℋ B → A 𝑀 ℋ * B

Proof

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