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 ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 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 ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑄 < 𝑋 )
6 breq1 ( 𝑃 = 𝑄 → ( 𝑃 < 𝑋𝑄 < 𝑋 ) )
7 5 6 syl5ibrcom ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( 𝑃 = 𝑄𝑃 < 𝑋 ) )
8 simp1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝐾 ∈ HL )
9 simp21 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑃𝐴 )
10 simp22 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑄𝐴 )
11 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
12 3 11 4 atlt ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑃𝑄 ) )
13 8 9 10 12 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑃𝑄 ) )
14 simp3l ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑃 𝑋 )
15 simp23 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑋𝐵 )
16 8 10 15 3jca ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵 ) )
17 2 3 pltle ( ( 𝐾 ∈ HL ∧ 𝑄𝐴𝑋𝐵 ) → ( 𝑄 < 𝑋𝑄 𝑋 ) )
18 16 5 17 sylc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑄 𝑋 )
19 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
20 19 3ad2ant1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝐾 ∈ Lat )
21 1 4 atbase ( 𝑃𝐴𝑃𝐵 )
22 9 21 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑃𝐵 )
23 1 4 atbase ( 𝑄𝐴𝑄𝐵 )
24 10 23 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑄𝐵 )
25 1 2 11 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑃𝐵𝑄𝐵𝑋𝐵 ) ) → ( ( 𝑃 𝑋𝑄 𝑋 ) ↔ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) 𝑋 ) )
26 20 22 24 15 25 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( ( 𝑃 𝑋𝑄 𝑋 ) ↔ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) 𝑋 ) )
27 14 18 26 mpbi2and ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) 𝑋 )
28 hlpos ( 𝐾 ∈ HL → 𝐾 ∈ Poset )
29 28 3ad2ant1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝐾 ∈ Poset )
30 1 11 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵 ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 )
31 20 22 24 30 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 )
32 1 2 3 pltletr ( ( 𝐾 ∈ Poset ∧ ( 𝑃𝐵 ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵𝑋𝐵 ) ) → ( ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) 𝑋 ) → 𝑃 < 𝑋 ) )
33 29 22 31 15 32 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) 𝑋 ) → 𝑃 < 𝑋 ) )
34 27 33 mpan2d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) → 𝑃 < 𝑋 ) )
35 13 34 sylbird ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → ( 𝑃𝑄𝑃 < 𝑋 ) )
36 7 35 pm2.61dne ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋𝑄 < 𝑋 ) ) → 𝑃 < 𝑋 )