Theorem dmdi

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

	Ref	Expression
Assertion	dmdi	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶 ) ) → ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dmdbr	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ ∀ 𝑥 ∈ Cℋ ( 𝐵 ⊆ 𝑥 → ( ( 𝑥 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝑥 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) ) )
2	1	biimpd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 → ∀ 𝑥 ∈ Cℋ ( 𝐵 ⊆ 𝑥 → ( ( 𝑥 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝑥 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) ) )
3		sseq2	⊢ ( 𝑥 = 𝐶 → ( 𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐶 ) )
4		ineq1	⊢ ( 𝑥 = 𝐶 → ( 𝑥 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐴 ) )
5	4	oveq1d	⊢ ( 𝑥 = 𝐶 → ( ( 𝑥 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) )
6		ineq1	⊢ ( 𝑥 = 𝐶 → ( 𝑥 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) )
7	5 6	eqeq12d	⊢ ( 𝑥 = 𝐶 → ( ( ( 𝑥 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝑥 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ↔ ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) )
8	3 7	imbi12d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐵 ⊆ 𝑥 → ( ( 𝑥 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝑥 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) ↔ ( 𝐵 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) ) )
9	8	rspcv	⊢ ( 𝐶 ∈ Cℋ → ( ∀ 𝑥 ∈ Cℋ ( 𝐵 ⊆ 𝑥 → ( ( 𝑥 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝑥 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( 𝐵 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) ) )
10	2 9	sylan9	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 → ( 𝐵 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) ) )
11	10	3impa	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 → ( 𝐵 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) ) )
12	11	imp32	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶 ) ) → ( ( 𝐶 ∩ 𝐴 ) ∨ℋ 𝐵 ) = ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) )