Metamath Proof Explorer


Theorem hlatj4

Description: Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 for atoms. (Contributed by NM, 9-Aug-2012)

Ref Expression
Hypotheses hlatjcom.j = ( join ‘ 𝐾 )
hlatjcom.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlatj4 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → ( ( 𝑃 𝑄 ) ( 𝑅 𝑆 ) ) = ( ( 𝑃 𝑅 ) ( 𝑄 𝑆 ) ) )

Proof

Step Hyp Ref Expression
1 hlatjcom.j = ( join ‘ 𝐾 )
2 hlatjcom.a 𝐴 = ( Atoms ‘ 𝐾 )
3 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
4 3 3ad2ant1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝐾 ∈ Lat )
5 simp2l ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑃𝐴 )
6 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 6 2 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
8 5 7 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
9 simp2r ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑄𝐴 )
10 6 2 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
11 9 10 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
12 simp3l ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑅𝐴 )
13 6 2 atbase ( 𝑅𝐴𝑅 ∈ ( Base ‘ 𝐾 ) )
14 12 13 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
15 simp3r ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑆𝐴 )
16 6 2 atbase ( 𝑆𝐴𝑆 ∈ ( Base ‘ 𝐾 ) )
17 15 16 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → 𝑆 ∈ ( Base ‘ 𝐾 ) )
18 6 1 latj4 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 𝑄 ) ( 𝑅 𝑆 ) ) = ( ( 𝑃 𝑅 ) ( 𝑄 𝑆 ) ) )
19 4 8 11 14 17 18 syl122anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑅𝐴𝑆𝐴 ) ) → ( ( 𝑃 𝑄 ) ( 𝑅 𝑆 ) ) = ( ( 𝑃 𝑅 ) ( 𝑄 𝑆 ) ) )