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Metamath Proof Explorer

Theorem chcv1

Description: The Hilbert lattice has the covering property. Proposition 1(ii) of Kalmbach p. 140 (and its converse). (Contributed by NM, 11-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chcv1 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 atom1d ⊢ ( 𝐵 ∈ HAtoms ↔ ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) )
2 spansncv2 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ ℋ ) → ( ¬ ( span ‘ { 𝑥 } ) ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ ( span ‘ { 𝑥 } ) ) ) )
3 sseq1 ⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐵 ⊆ 𝐴 ↔ ( span ‘ { 𝑥 } ) ⊆ 𝐴 ) )
4 3 notbid ⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ ¬ ( span ‘ { 𝑥 } ) ⊆ 𝐴 ) )
5 oveq2 ⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐴 ∨ℋ ( span ‘ { 𝑥 } ) ) )
6 5 breq2d ⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ ( span ‘ { 𝑥 } ) ) ) )
7 4 6 imbi12d ⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( ( ¬ 𝐵 ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) ↔ ( ¬ ( span ‘ { 𝑥 } ) ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ ( span ‘ { 𝑥 } ) ) ) ) )
8 2 7 syl5ibrcom ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ ℋ ) → ( 𝐵 = ( span ‘ { 𝑥 } ) → ( ¬ 𝐵 ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) ) )
9 8 adantld ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ ℋ ) → ( ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) → ( ¬ 𝐵 ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) ) )
10 9 rexlimdva ⊢ ( 𝐴 ∈ Cℋ → ( ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) → ( ¬ 𝐵 ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) ) )
11 10 imp ⊢ ( ( 𝐴 ∈ Cℋ ∧ ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) ) → ( ¬ 𝐵 ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )
12 1 11 sylan2b ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 → 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )
13 atelch ⊢ ( 𝐵 ∈ HAtoms → 𝐵 ∈ Cℋ )
14 chjcl ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )
15 cvpss ⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ ) → ( 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) → 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ) )
16 14 15 syldan ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) → 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ) )
17 chnle ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ) )
18 16 17 sylibrd ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) → ¬ 𝐵 ⊆ 𝐴 ) )
19 13 18 sylan2 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) → ¬ 𝐵 ⊆ 𝐴 ) )
20 12 19 impbid ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )