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 ( ( 𝐴C𝐵 ∈ HAtoms ) → ( ¬ 𝐵𝐴𝐴 ( 𝐴 𝐵 ) ) )

Proof

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