Metamath Proof Explorer


Theorem hlrelat

Description: A Hilbert lattice is relatively atomic. Remark 2 of Kalmbach p. 149. ( chrelati analog.) (Contributed by NM, 4-Feb-2012)

Ref Expression
Hypotheses hlrelat5.b 𝐵 = ( Base ‘ 𝐾 )
hlrelat5.l = ( le ‘ 𝐾 )
hlrelat5.s < = ( lt ‘ 𝐾 )
hlrelat5.j = ( join ‘ 𝐾 )
hlrelat5.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlrelat ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝𝐴 ( 𝑋 < ( 𝑋 𝑝 ) ∧ ( 𝑋 𝑝 ) 𝑌 ) )

Proof

Step Hyp Ref Expression
1 hlrelat5.b 𝐵 = ( Base ‘ 𝐾 )
2 hlrelat5.l = ( le ‘ 𝐾 )
3 hlrelat5.s < = ( lt ‘ 𝐾 )
4 hlrelat5.j = ( join ‘ 𝐾 )
5 hlrelat5.a 𝐴 = ( Atoms ‘ 𝐾 )
6 1 2 3 5 hlrelat1 ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 < 𝑌 → ∃ 𝑝𝐴 ( ¬ 𝑝 𝑋𝑝 𝑌 ) ) )
7 6 imp ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝𝐴 ( ¬ 𝑝 𝑋𝑝 𝑌 ) )
8 simpll1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → 𝐾 ∈ HL )
9 8 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → 𝐾 ∈ Lat )
10 simpll2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → 𝑋𝐵 )
11 1 5 atbase ( 𝑝𝐴𝑝𝐵 )
12 11 adantl ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → 𝑝𝐵 )
13 1 2 3 4 latnle ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑝𝐵 ) → ( ¬ 𝑝 𝑋𝑋 < ( 𝑋 𝑝 ) ) )
14 9 10 12 13 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → ( ¬ 𝑝 𝑋𝑋 < ( 𝑋 𝑝 ) ) )
15 2 3 pltle ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 < 𝑌𝑋 𝑌 ) )
16 15 imp ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) → 𝑋 𝑌 )
17 16 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → 𝑋 𝑌 )
18 17 biantrurd ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → ( 𝑝 𝑌 ↔ ( 𝑋 𝑌𝑝 𝑌 ) ) )
19 simpll3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → 𝑌𝐵 )
20 1 2 4 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑝𝐵𝑌𝐵 ) ) → ( ( 𝑋 𝑌𝑝 𝑌 ) ↔ ( 𝑋 𝑝 ) 𝑌 ) )
21 9 10 12 19 20 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → ( ( 𝑋 𝑌𝑝 𝑌 ) ↔ ( 𝑋 𝑝 ) 𝑌 ) )
22 18 21 bitrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → ( 𝑝 𝑌 ↔ ( 𝑋 𝑝 ) 𝑌 ) )
23 14 22 anbi12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝𝐴 ) → ( ( ¬ 𝑝 𝑋𝑝 𝑌 ) ↔ ( 𝑋 < ( 𝑋 𝑝 ) ∧ ( 𝑋 𝑝 ) 𝑌 ) ) )
24 23 rexbidva ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) → ( ∃ 𝑝𝐴 ( ¬ 𝑝 𝑋𝑝 𝑌 ) ↔ ∃ 𝑝𝐴 ( 𝑋 < ( 𝑋 𝑝 ) ∧ ( 𝑋 𝑝 ) 𝑌 ) ) )
25 7 24 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝𝐴 ( 𝑋 < ( 𝑋 𝑝 ) ∧ ( 𝑋 𝑝 ) 𝑌 ) )