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Metamath Proof Explorer

Theorem dmdsl3

Description: Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of MaedaMaeda p. 2. (Contributed by NM, 26-Apr-2006) (New usage is discouraged.)

Ref Expression
Assertion dmdsl3 ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = 𝐶 )

Proof

Step Hyp Ref Expression
1 dmdi ⊢ ( ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
2 1 exp32 ⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐵 𝑀ℋ* 𝐴 → ( 𝐴 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) ) ) )
3 2 3com12 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐵 𝑀ℋ* 𝐴 → ( 𝐴 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) ) ) )
4 3 imp32 ⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
5 4 3adantr3 ⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
6 chjcom ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
7 6 ineq2d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
8 7 3adant3 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
9 dfss2 ⊢ ( 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ↔ ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = 𝐶 )
10 9 biimpi ⊢ ( 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) → ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = 𝐶 )
11 8 10 sylan9req ⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = 𝐶 )
12 11 3ad2antr3 ⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = 𝐶 )
13 5 12 eqtrd ⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = 𝐶 )