Metamath Proof Explorer

Theorem atlelt

Description: Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012)

Ref Expression
Hypotheses atlelt.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
atlelt.l ⊢ ≤ = ( le ‘ 𝐾 )
atlelt.s ⊢ < = ( lt ‘ 𝐾 )
atlelt.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atlelt ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 < 𝑋 )

Proof

Step Hyp Ref Expression
1 atlelt.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 atlelt.l ⊢ ≤ = ( le ‘ 𝐾 )
3 atlelt.s ⊢ < = ( lt ‘ 𝐾 )
4 atlelt.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 simp3r ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 < 𝑋 )
6 breq1 ⊢ ( 𝑃 = 𝑄 → ( 𝑃 < 𝑋 ↔ 𝑄 < 𝑋 ) )
7 5 6 syl5ibrcom ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 = 𝑄 → 𝑃 < 𝑋 ) )
8 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝐾 ∈ HL )
9 simp21 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 ∈ 𝐴 )
10 simp22 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 ∈ 𝐴 )
11 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
12 3 11 4 atlt ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑃 ≠ 𝑄 ) )
13 8 9 10 12 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑃 ≠ 𝑄 ) )
14 simp3l ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 ≤ 𝑋 )
15 simp23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑋 ∈ 𝐵 )
16 8 10 15 3jca ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) )
17 2 3 pltle ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑄 < 𝑋 → 𝑄 ≤ 𝑋 ) )
18 16 5 17 sylc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 ≤ 𝑋 )
19 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
20 19 3ad2ant1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝐾 ∈ Lat )
21 1 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
22 9 21 syl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 ∈ 𝐵 )
23 1 4 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
24 10 23 syl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 ∈ 𝐵 )
25 1 2 11 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋 ) ↔ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) )
26 20 22 24 15 25 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( ( 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋 ) ↔ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) )
27 14 18 26 mpbi2and ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 )
28 hlpos ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Poset )
29 28 3ad2ant1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝐾 ∈ Poset )
30 1 11 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 )
31 20 22 24 30 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 )
32 1 2 3 pltletr ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑃 ∈ 𝐵 ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) → 𝑃 < 𝑋 ) )
33 29 22 31 15 32 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) → 𝑃 < 𝑋 ) )
34 27 33 mpan2d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) → 𝑃 < 𝑋 ) )
35 13 34 sylbird ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 ≠ 𝑄 → 𝑃 < 𝑋 ) )
36 7 35 pm2.61dne ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 < 𝑋 )