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Metamath Proof Explorer

Theorem hlrelat5N

Description: An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of Kalmbach p. 149. (Contributed by NM, 21-Oct-2011) (New usage is discouraged.)

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 hlrelat5N ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ 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 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
9 id ⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵 )
10 1 5 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
11 ovexd ⊢ ( 𝑝 ∈ 𝐵 → ( 𝑋 ∨ 𝑝 ) ∈ V )
12 2 3 pltval ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑝 ) ∈ V ) → ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ↔ ( 𝑋 ≤ ( 𝑋 ∨ 𝑝 ) ∧ 𝑋 ≠ ( 𝑋 ∨ 𝑝 ) ) ) )
13 11 12 syl3an3 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ↔ ( 𝑋 ≤ ( 𝑋 ∨ 𝑝 ) ∧ 𝑋 ≠ ( 𝑋 ∨ 𝑝 ) ) ) )
14 1 2 4 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → 𝑋 ≤ ( 𝑋 ∨ 𝑝 ) )
15 14 biantrurd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑋 ≠ ( 𝑋 ∨ 𝑝 ) ↔ ( 𝑋 ≤ ( 𝑋 ∨ 𝑝 ) ∧ 𝑋 ≠ ( 𝑋 ∨ 𝑝 ) ) ) )
16 13 15 bitr4d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ↔ 𝑋 ≠ ( 𝑋 ∨ 𝑝 ) ) )
17 1 2 4 latleeqj1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑝 ≤ 𝑋 ↔ ( 𝑝 ∨ 𝑋 ) = 𝑋 ) )
18 17 3com23 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑝 ≤ 𝑋 ↔ ( 𝑝 ∨ 𝑋 ) = 𝑋 ) )
19 1 4 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑝 ) = ( 𝑝 ∨ 𝑋 ) )
20 19 eqeq1d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( ( 𝑋 ∨ 𝑝 ) = 𝑋 ↔ ( 𝑝 ∨ 𝑋 ) = 𝑋 ) )
21 18 20 bitr4d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑝 ≤ 𝑋 ↔ ( 𝑋 ∨ 𝑝 ) = 𝑋 ) )
22 21 notbid ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( ¬ 𝑝 ≤ 𝑋 ↔ ¬ ( 𝑋 ∨ 𝑝 ) = 𝑋 ) )
23 nesym ⊢ ( 𝑋 ≠ ( 𝑋 ∨ 𝑝 ) ↔ ¬ ( 𝑋 ∨ 𝑝 ) = 𝑋 )
24 22 23 bitr4di ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( ¬ 𝑝 ≤ 𝑋 ↔ 𝑋 ≠ ( 𝑋 ∨ 𝑝 ) ) )
25 16 24 bitr4d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ↔ ¬ 𝑝 ≤ 𝑋 ) )
26 8 9 10 25 syl3an ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐴 ) → ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ↔ ¬ 𝑝 ≤ 𝑋 ) )
27 26 3expa ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ↔ ¬ 𝑝 ≤ 𝑋 ) )
28 27 anbi1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ 𝑝 ≤ 𝑌 ) ↔ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )
29 28 rexbidva ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ 𝑝 ≤ 𝑌 ) ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )
30 29 3adant3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ 𝑝 ≤ 𝑌 ) ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )
31 30 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ( ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ 𝑝 ≤ 𝑌 ) ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )
32 7 31 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ 𝑝 ≤ 𝑌 ) )