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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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )
7 6 imp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) )
8 simpll1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ HL )
9 8 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ Lat )
10 simpll2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
11 1 5 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
12 11 adantl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑝 ∈ 𝐵 )
13 1 2 3 4 latnle ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( ¬ 𝑝 ≤ 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑝 ) ) )
14 9 10 12 13 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( ¬ 𝑝 ≤ 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑝 ) ) )
15 2 3 pltle ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → 𝑋 ≤ 𝑌 ) )
16 15 imp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → 𝑋 ≤ 𝑌 )
17 16 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ≤ 𝑌 )
18 17 biantrurd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ) )
19 simpll3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 )
20 1 2 4 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) )
21 9 10 12 19 20 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) )
22 18 21 bitrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) )
23 14 22 anbi12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) )
24 23 rexbidva ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ( ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ↔ ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) )
25 7 24 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) )